THE 

ELEMENTS 

OF THE 

DIFFERENTIAL CALCULUS 

COMPREHENDING THE 

GENERAL THEORY OF CURVE SURFACES. 

AND OF 

CURVES OF DOUBLE CURVATURE. 

INTENDED FOR THE USE OF 
MATHEMATICAL STUDENTS IN SCHOOLS AND UNIVERSITIES. 

BY J; R. YOUNG, 

AUTHOR OF 
"THE ELEMENTS OF ANALYTICAL GEOMETRY." 



Rp;V!SED AND CORRECTED, BY 

MICHAEL O'SHANNESSY, A.M. 



PHILADELPHIA: 
HOGAN AND THOMPSON, 

No. 30 North Fourth Street. 



1839. 









Entered according to Act of Congress, the 9th of May, in the year 1833, by 
Carey, Lea &. Blanchard, in the oifice of the Clerk of the Southern District of 
New York. 



C. Sherman & Co. Printers, Philadelphia. 



ADVERTISEMENT. 



This edition of Young's Differential and Integral Calculus 
is presented to the American public, with a confidence in its 
favourable reception, proportionate to that which the original 
acquired in England. The text has not been materially al- 
tered, though many errors have been corrected, some of which 
by Professor Dodd of Princeton College, N. J. 

These volumes will be found to contain a full elementary 
course of the subject of which they treat, and well adapted as 
a text book for Colleges and Universities. 

The second volume, treating exclusively of the Integral 
Calculus, is now in press, and will be speedily published. 

New- York, March, 1833. 



PREFACE 



The object of the present volume is to teach the principles of the 
Differential Calculus, and to show the application of these principles 
to several interesting and important inquiries, more particularly to the 
general theory of Curves and Surfaces. Throughout these applica- 
tions I have endeavoured to preserve the strictest rigour in the various 
processes employed, so that the student who may have hitherto been 
accustomed only to the pure reasoning of the ancient geometry will 
not, I think, find in these higher order of researches any principle 
adopted, or any assumption made, inconsistent with his previous no- 
tions of mathematical accuracy. If I have, indeed, succeeded in 
accomplishing this very desirable object, and have really shown 
that the applications of the Calculus do not necessarily involve any 
principle that will not bear the most scrupulous examination, I may, 
perhaps, be allowed to think that I have, in this small volume, con- 
tributed a little towards the perfecting of the most powerful instru- 
ment which the modern analysis places in the hand of the mathema- 
tician. 

It is the adoption of exceptionable principles, and even, in some 
cases, of contradictory theories, into the elements of this science, 
that have no doubt been the chief causes why it has hitherto been so 
little studied in a country where the ancient geometry has been so 
extensively and so successfully cultivated. The student who pro- 
ceeds from the works of Euclid or of Jlpollonius to study those of our 
modern analysts, will be naturally enough startled to find that in the 
theory of the Differential Calculus he is to consider that as absolutely 
nothing which, in the application of that theory, is to be considered 
a quantity infinitely small. He will naturally enough be startled to 
find that a conclusion is to be taken as general, when he is at the 



VI PREFACE. 

same time told that the process which led to that conclusion has fail- 
ing cases ; and yet one or both of these inconsistencies pervade more 
or less every book on the Calculus which I have had an opportunity 
of examining. 

The whole theory of what the French mathematicians vaguely call 
consecutive points and consecutive elements, involves the first of these 
objectionable principles ;* for, if the abscissa of any point be repre- 
sented by x, then the abscissa of the consecutive point, or that sepa- 
rated from the former by an infinitely small interval, is represented 
by x + dx, although dx, at the outset of the subject, is said to be 0. 
Again, the theory of tangents, the radius of curvature, principles of 
osculation, &c, are all made to depend upon Taylor's theorem, and 
therefore can strictly apply only at those points of the curve where 
this theorem does not fail : the conclusions, however, are to be re- 
ceived in all their generality, f 

* It is to be regretted that terms so vague and indefinite should be introduced 
into the exact sciences ; and it is more to be regretted that English elementary writers 
should adopt them merely because they are used by the French, and that too with- 
out examining into the import these terms carry in the works from which they are 
copied. In a recent production of the University of Cambridge, the author, in at- 
tempting to follow the French mode of solving a certain problem, has confounded 
consecutive points with consecutive elements, two very distinct things: although 
neither very intelligible, the consequence of this mistake is, that the result is not 
what was intended; so that, after the process is fairly finished, a new counter- 
balancing error is introduced, and thus the solution righted ! 

1 1 am anxious not to be misunderstood here, and shall therefore state specifi- 
cally the nature of my objection. In establishing the theory of contact, &c, by 
aid of Taylor's theorem, it is assumed that a value may be given to the increment 
h so small as to render the term into which it enters greater than all the following 
terms of the series taken together. Now how can a function of absolutely inde- 
terminate quantities be shown to be greater or less than a series of other functions 
of the same indeterminate quantities without, at least, assuming some determinate 
relation among them ? If we say that the assertion applies, whatever particular 
value we substitute for the indeterminate in the proposed functions or differential 
coefficients, we merely shift the dilemma, for an indefinite number of these particu- 
lar values may render the functions all infinite ; and we shall be equally at a loss 
to conceive how one of these infinite quantities can be greater or less than the 
others. It appears, therefore, that the usual process by which the theory of con- 
tact is established, applies rigorously only to those points of curves for which 
Taylor's development does not fail, and I cannot help thinking that on these 
grounds the Analytical Theory of Functions, by Lagrange, in its application to Ge- 



PREFACE. Vll 

If this statement be true, it is not to be wondered at that students 
so often abandon the study of this science, less discouraged with its 
difficulties than disgusted with its inconsistencies. To remove these 
inconsistencies, which so often harass and impede the student's pro- 
gress, has been my object in the present volume ; and, although my 
endeavours may not have entirely succeeded, I have still reason to 
hope that they have not entirely failed. The following brief outline 
will convey a notion of the extent and pretensions of the book ; a more 
detailed enumeration of the various topics treated of, will be found in 
the table of contents. 

I have taken for the basis of the theory the method of limits first 
employed by Newton, although designated by foreign writers as the 
method of oVMemhert. I consider this method to be as unexceptiona- 
ble as that of Lagrange, and on account of its greater simplicity, 
better adapted to elementary instruction. 

The First Chapter is devoted to the exposition of the fundamental 
principles ; and in explaining the notation I have been careful to im- 
press upon the student's mind that the differentials dx, dy, &c. are in 
themselves absolutely of no value, and that their ratios only are sig- 
nificant : this is the foundation of the whole theory, and it has been 
adhered to throughout the volume, without any shifting of the hypo- 
thesis. 

In the Second Chapter it is shown, that if fx represent any function 
of x, and x be changed into x + h, the new state j {x + h) of the 
function- may always be developed according to the ascending inte- 
gral powers of the increment h ; and this leads to the important con- 
clusion that the coefficient of the second term in the development of 
the function f(x + h) is the differential coefficient derived from the 
function /#; a fact which Lagrange has made the foundation of his 

ometry is defective, although I feel anxious to express my opinion of that celebra- 
ted performance with all becoming caution and humility. Indeed Lagrange him- 
self has admitted this defect, and observes, (Thiorie des Fonctions, p. 181,) " duoi- 
que ces exceptions ne portent aucune atteinte a la theorie generale, il est neces- 
saire, pour ne rien laisser a desirer, de voir comment elle doit etre modifier dans 
les cas particuliers dont il s'agit." (See note C at the end.) But he has not 
modified the expression deduced from this exceptionable theory for the radius of 
curvature, which indeed is always applicable whether the differential coefficients 
become infinite or not, although, for reasons already assigned, the process which 
led to it restricts its application to particular points. 



Vlll PREFACE. 

theory of analytical functions. The chapter then goes on to treat of 
the differentiation of the various kinds of functions, algebraic and 
transcendental, direct and inverse, and concludes with an article on 
successive differentiation. 

The Third Chapter is devoted to Maclaurin?s theorem, and its ap- 
plication is shown in the development of a great variety of functions. 
Occasion is taken, in the course of this chapter, to introduce to the 
student's attention some valuable analytical formulas and expressions 
from Eider, Demoivre, Cotes, and other celebrated analysts, together 
with those curious properties of the circle discovered by Co tes and 
Demoivre. 

The Fourth Chapter is on Taylor's theorem, which makes known 
the actual development of the function f(x + h) according to the 
form established in the second chapter. From this theorem are de- 
rived commodious expressions for the total differential coefficient 
when the function is complicated, and whether its form be explicit or 
implicit ; the whole being illustrated by a variety of examples. 

The Fifth Chapter contains the complete theory of vanishing frac- 
tions. 

The Sixth is on the maxima and minima values of functions of a 
single variable, and will, I think, be found to contain several original 
remarks and improved processes. 

Chapter the Seventh is on the differentiation and development of 
functions of two independent variables. The usual method of obtain- 
ing the development of a function of two variables according to the 
powers of the increments, is to develop first on the supposition that x 
only varies and that y is constant, and afterwards to consider y, which 
is assumed to enter into the coefficients, to be changed into y + h. 
But y may be so combined with x in the function F (x, y) that it shall, 
when considered as a constant, disappear from all the differential co- 
efficients, which circumstances should be pointed out and be shown 
not to affect the truth of the result : I have, however, avoided the ne- 
cessity of showing this, by proceeding rather differently. The chap- 
ter concludes with Lagrange's Theorem, concisely demonstrated and 
applied to several examples. 

The Eighth Chapter completes the theory of maxima and minima, 
by applying the principles delivered in chapter VI. to functions of two 
independent variables, and it also contains an important article on 



PREFACE. IX 

changing the independent variable, a subject very improperly omitted 
in all the English books. 

The Ninth Chapter is devoted to a matter of considerable import- 
ance, viz. to the examination of the cases in which Taylor's theorem 
fails ; and I have, I think, satisfactorily shown, that these failing cases 
are always indicated by the differential coefficients becoming infinite, 
and that the theorem does not fail when these coefficients become 
imaginary, as Lacroix, and others after him, have asserted. Besides 
the correction of this erroneous doctrine, which has been sanctioned 
by names of the highest reputation, another very remarkable over- 
sight, though of far less importance, is detected in the Calcul des 
Fonctions of Lagrange, and is pointed out in the present chapter : it 
has been unsuspectingly copied by other writers ; and thus an entirely 
wrong solution to a very simple problem has been printed, and re- 
printed, without any examination into the principles employed in it ; 
and which, I suppose, the high reputation of Lagrange was consider- 
ed to render unnecessary. 

These nine chapters constitute the First Section of the work, and 
comprise the pure theory of the subject ; the remaining part is devot- 
ed to the application of this to geometry, and is divided into two parts, 
the first containing the theory of plane curves, and the second the 
theory of curve surfaces, and of curves of double curvature. 

The First Chapter in the Second Section explains the method of tan- 
gents, and the general differential equation of the tangent to any plane 
curve is obtained by the same means that the equation is obtained in 
analytical geometry, and is therefore independent of the failing cases of 
Taylor's theorem. The method of tangents naturally leads to the con- 
sideration of rectilinear asymptotes, which is, therefore, treated of in this 
chapter, and several examples are given, as well when the curve is 
referred to polar as to rectangular coordinates, and a few passing ob- 
servations made on the circular asymptotes to spiral curves, the chap- 
ter terminating with the differential expression for the arc of any plane 
curve determined without the aid of Taylor's theorem. 

The Second Chapter contains the theory of osculation, which is 
shown to be unaffected by the failing cases of Taylor's theorem, al- 
though this is employed to establish the theory. The expressions for 
the radius of curvature are afterwards deduced, and several examples 

B 



X PREFACE. 

of their application given principally to the curves of the second order, 
and an instance of their utility shown in determining the ratio of the 
earth's diameters. 

The Third Chapter is on involutes, evolutes, and consecutive curves, 
and contains some interesting theorems and practical examples. Of 
what the French call consecutive curves, I have endeavoured to give 
a clear and satisfactory explanation, unmixed with any vague notions 
about infinity. 

The Fourth Chapter is on the singular points of curves, and con- 
tains easy rules for detecting them, from an examination of the equa- 
tion of the curve. This chapter also contains the general theory of 
curvilinear asymptotes, and completes the Second Section, or that 
assigned to the consideration of plane curves. 

The Third Section is devoted to the general theory of curve sur- 
faces, and of curves of double curvature ; in the First Chapter of 
which are established the several forms of the equations of the tan- 
gent plane and normal line at any point of a curve surface, and of the 
linear tangent and normal plane at any point of a curve of double cur- 
vature. 

In the Second Chapter the theory of conical and cylindrical surfa- 
ces is discussed, as also that of surfaces of revolution ; and that re- 
markable case is examined, where the revolution of a straight line 
produces the same surface as the revolution of the hyperbola, to which 
this line is an asymptote. Throughout this chapter are interspersed 
many valuable and interesting applications of the calculus, chiefly 
from Monge. The Third Chapter embraces the theory of the curva- 
ture of surfaces in general, and will be found to form a collection of 
very beautiful theorems, the results, principally, of the researches of 
Euler, Monge, and Dupin. Most of these theorems have, however, 
usually been established by the aid of the infinitesimal calculus, or by 
the use of some other equally objectionable principle ; they are here 
fairly deduced from the principles of the differential calculus, without, 
in any instance, departing from those principles, as laid down in the 
preliminary chapter. Those who are familiar with these inquiries will 
find that I have obtained some of these theorems in a manner much 
more simple and concise than has hitherto been done. I need only 
mention here, as instances of this simplicity, the theorems of Euler 
and of Meusnier, at pages 182 and 186. 



PREFACE. Xi 

The Fourth Chapter is on twisted surfaces, a class of surfaces which 
have never been treated of, to any extent, by any English author, al- 
though, as has been recently shown, the English were the first who 
noticed the peculiarities of certain individual surfaces belonging to 
this extensive class.* For what is here given, I am indebted to the 
French mathematicians, to JWonge principally, and also to the Che~ 
valier Le Roy, who has recently published a very neat and compre- 
hensive little treatise on curves and surfaces. 

The Fifth Chapter treats on the developable surfaces, or those 
which, like the cone and cylinder, may, if flexible, be unrolled upon 
a plane, without being twisted or torn. The Sixth Chapter is on 
curves of double curvature ; and the Seventh, which concludes the 
volume, contains a few miscellaneous propositions intimately connect- 
ed with the theory of surfaces. From the foregoing brief analysis, it 
will appear evident to those familiar with the present state of mathe- 
matical instruction in this country, that I have introduced, into a little 
duodecimo volume, a more comprehensive view of the theory and 
applications of the differential calculus than has yet appeared in the 
English language. But I have aimed at more than this ; I have en- 
deavoured to simplify and improve much that I have adopted from 
foreign sources ; and, above all, to establish every thing here taught, 
upon principles free from inconsistency and logical objections ; and 
if it be found, upon examination, that 1 have entirely failed in this en- 
deavour, I shall certainly feel a proportionate disappointment. 

I am. not, however, so sanguine as to look for much public en- 
couragement of my labours, however successfully they may have 
been devoted : it is not customary to place much value, in this coun- 
try, upon any mathematical production, of whatever merit, that does 
not emanate from Cambridge. The hereditary reputation enjoyed 
by this University, and bequeathed to it by the genius of Barrow, of 
Neivton, and of Cotes, seems to have endowed it with such strong 
claims on the public attention and respect, that every thing it puts 
forth is always received as the best of its kind. If this be the case 
with Cambridge books, of course it is also the case with Cambridge 
men, and accordingly we find almost all our public mathematical 
situations filled by members of this University. It is true that now 

* See Leyboxirn 1 s Repository, No. 22. 



Xll PREFACE. 

and then, in the course of half a century, we find an exception to this ; 
one or two instances on record have undoubtedly occurred, where it 
has been, by some means or other, discovered that men who had ne- 
ver seen Cambridge knew a little of mathematics, as in the case of 
Thomas Simpson, and of Dr. Hutton ; but such instances are rare. 
It is not for me to inquire into the justice of this exclusive system ; 
but, while such a system prevails, there need be little wonder at the 
decline of science in England : while all inducement to cultivate sci- 
ence is thus confined to a particular set of men, no wonder that its 
votaries are few. It is to be hoped, however, that in the present 
" liberal and enlightened age," such a state of things will not long 
continue, and that even the poor and unfriended student may be cheer- 
ed up, amidst all the obstacles that surround him, in the laborious and 
difficult, but sublime and elevating career on which he has entered, 
by a well-founded assurance that his exertions, if successful, will not 
be the less appreciated because they were solitary and unassisted. 

May 12, 1831. 

J. R. YOUNG. 



CONTENTS 



SECTION I. 

On the Differentiation of Functions in general. 
Article Page 

1. A Function defined -_----- 1 

2. Effect produced on the function by a change in the variable - 2 

3. Differential coefficient determined - - - - 3 

4. General form of the development of/ (x + h) - - 5 

5. The coefficient of the second term in the general development is the dif- 

ferential coefficient derived from the function fx - - 8 

6. To differentiate the product of two or more functions of the same variable 9 

7. To differentiate a fraction - - - - - -10 

8. To differentiate any power of a function - - - - ib. 

9. To differentiate an expression consisting of several functions of the same 

variable - - - - - - - -12 

10. Application of the preceding rules to examples - - - ib. 

11. Transcendental functions - - - - - - 15 

12. To find the differential of a logarithm - - - - ib. 

13. To differentiate an exponential function - - - - 16 

Examples on transcendental functions - - - - ib. 

14. To differentiate circular functions - - - - - 19 

15. Differentiation of inverse functions - - - - - 21 

16. Forms of the differentials when the radius is arbitrary - - - 24 

17. Successive differentiation explained - - - - - 25 

18. Illustrations of the process - - - - - - 26 

19. Investigation ofMaclauriri's Theorem - - - - - 28 

20. Application of Maclaurin's theorem to the development of functions - 29 

21. Deduction of Euler's expressions for the sine and cosine of an arc, by 

means of imaginary exponentials - - - - - 31 

22. Demoivre's formula, and series for the sine and cosine of a multiple arc - 32 

23. Decomposition of the expression y 2m — 2y cos. 8 -{- 1 into its quadratic 

factors - - - - - -- -33 

24. Demoivre's property of the circle - - - - - 34 

25. Cotes' s properties of the circle - - - - - .35 



XIV CONTENTS. 

Article Page 

*/ri 

26. John BarnoulWs development of ( \/ — 1) - - - ib. 

Developments of tan. x and tan. ~ l x - - - - 36 

27. Ruler's series for approximating to the circumference of a circle - 38 

28. Bertrand's more convergent series - - - - - ib. 

29. Examples for exercise - - - - - - -39 

30. Investigation of Taylor's Theorem - - - - - 40 

31. MaclauHn's theorem deduced from Taylor's - - - - 42 

32. Application of Taylor's theorem to the development of functions - ib. 

33. Of a function of a function of a single variable - - - - 44 

34. Examples of the application of this form - - - - 45 

35. Form of the differential coefficient derived from the function u = F(p,q,) 

where p and q are functions of the same variable - - - ib. 

36. Form of the coefficient when the function is v = F (p, q, r,) - - 46 

37. Distinction between partial and total differential coefficients - - 47 

38. Examples -------- 48 

39. Differentiation and development of implicit functions - - - 49 

40. On vanishing fractions - - - - - - -52 

41. Application of the calculus to determine the true value of a vanishing 

fraction - - - - - - - -53 

42. Determination of the value when Taylor's theorem fails - - 56 

43. Determination of the value of a fraction, of which both numerator and de- 

nominator are infinite - - - - - - 59 

44. Determination of the value of the product of two factors, when one be- 

comes and the other oo- - - - --60 

45. Determination of the value of the difference of two functions, when they 

both become infinite - - - - - - ib # 

46. Examples on the preceding theory - - - - - 61 

47. On the maxima and minima values of functions of a single variable - 63 

48. If the function F (a -f- h) be developed according to the ascending pow- 

ers of h, a value so small may be given to h that any proposed term 

in the series shall exceed the sum of all that follow - - 64 

49. Determination of the maxima and minima values in those cases where 

Taylor's theorem is applicable - .- - - - ib. 

50. Determination of the values when Taylor's theorem is not applicable - 66 

51. Maxima and minima values which satisfy the condition — - = oo - 68 

ax 

52. Conditions of maxima and minima, when the function is implicitly given ib. 

53. Precepts to abridge the process of finding maxima and minima values 69 

54. Examples -------. 70 

55. On the cautions to be observed in applying the analytical theory of maxi- 

ma and minima to Geometry - - - - - 79 

56. Differentiation of functions of two independent variables - - 81 

57. Form of the differential when the function is implicit - 82 



CONTENTS. XV 

Article Page 

58. The ratio of the two partial differential coefficients derived from u == Fz, 

2 being a function of x and y, is independent of F - - 83 

59. Development of functions of two independent variables - - 84 

60. The partial differential coefficients composing the coefficient of any term 

in the general development are identical with those arising from dif- 
ferentiating the preceding term - - - - - 87 

61. Maclaurin's theorem extended to functions of two independent variables 88 

62. Lagrange's Theorem - - - - - - -89 

63. Applications of Lagrange's theorem - - - - - 91 

64. Maxima and minima values of functions of two variables - - 94 

65. Examples - - - - - - - -96 

66. On changing the independent variable - - - 99 

67. On the failing cases of Taylor's Theorem - - - - 100 

68. Explanation of the cause and extent of these failing cases - - ib # 

69. Particular examination of them - 102 

70. Inferences from this examination - 103 

71. The converse of these inferences true ----- 104 

72. To obtain the true development when Taylor's theorem fails - - ib. 

73. Correction of the errors of some analysts with respect to the failing cases 

of Taylor's theorem .-..-_ 106 

74. On the multiple values of — in implicit functions - 108 

dx 

75. Determination of these multiple values - 109 

76. Determination of the multiple values of —4 - - - - 111 

dx % 



SECTION II. 

Application of the Differential Calculus to the Theory of Plane Curves. 

77. Explanation of the method of tangents - - - - 113 

78. Equation of the normal - - - - - -114 

79. Application to curves related to rectangular coordinates - -116 

80. Formulas for polar curves - - - - - -117 

81. Application to spirals - - -- - - -119 

82. Rectilinear asymptotes - - - - - -120 

83. Examples on the determination of asymptotes - - - 122 

84. Rectilinear asymptotes to spirals - 123 

85. Circular asymptotes to spirals ■ 124 

86. Expression for the differential of an arc of a plane curve . - 125 

87. Principles of osculation - - - - - -126 

88. Different orders of contact - - - - - - 128 



XVI CONTENTS. 

Article Pagt 

89. Nature of the contact at those points for which Taylor's development 

holds • -129 

90. Of the contact at the points for which Taylor's development fails - ib. 

91. Osculating circle ------- 130 

92. Determination of the radius of curvature - - - - ib. 

93. The centres of touching circles all on the normal - 132 

94. Examples on the radius of curvature ----- ib. 

95. Expression for the radius of curvature of an ellipse applied to determining 

the ratio of the polar and equatorial diameters of the earth - - 134 

96. To determine those points in a given curve, at which the osculating circle 

shall have contact of the third order - 136 

97. Expression for the radius of curvature when the independent variable is 

arbitrary ----- - - ib. 

98. Particular forms derivable from this general expression - - 137 

99. Suitable formula for polar curves - - - - - 138 

100. Involutes and evolutes -.-.-. 140 

101. Determination of the evolutes of several curves - 141 

102. Normals to the involute are tangents to the evolute ... 143 

103. The difference of any two radii of curvature is equal to the arc of the 

evolute comprehended between them - - - - ib # 

104. On consecutive lines and curves - - - - . 145 

105. Determination of the points of intersection of consecutive curves - ib. 

106. Determination of the envelope of a family of curves - - -146 

107. Examples of this theory ----__ 147 

108. Multiple points of curves - - - - - -150 

109. Determination of these points from the equation of the curve - - 151 

110. Conjugate points -- - - - - -152 

111. The determination of these point does not depend on Taylor's theorem ib. 

112. Multiple points of the second species - 153 

113. Cusps or points of regression - - - - - -154 

114. Cusps exist only at points for which Taylor's theorem fails - - ib. 

115. To distinguish a limit from a cusp - - - - - ib. 

116. Examples of the determination of cusps whose branches touch an ordi- 

nate or an abscissa - - - - - - -155 

117. Cusps whose branches touch a line oblique to the axes - - 156 

118. Conditions fulfilled by such cusps ----- 157 

119. Distinction between cusps of the first and those of the second kind - ib. 

120. Examples - - - - - - - -ib. 

121. On points of inflexion ------- 158 

122. On curvilinear asymptotes - - - - - -161 



CONTENTS. XV11 

SECTION III. 

On the general Theory of Curve Surfaces and of Curves of Double Curvature. 

Article Page 

123. To determine the equation of the tangent plane at any point on a curve 

surface -__----- 164 

124. Form of the equation when the equation of the surface is implicit - 165 

125. To determine the equation of the normal line at any point of a curve 

surface - - - - - - - -ib. 

126. Expressions for the inclinations of the normal to the axes - - 166 

127. Forms of these expressions when the equation of the surface is implicit ib. 

128. To determine the equation of the linear tangent at any point of a curve 

of double curvature - - - - - - - ib. 

129. To determine the equation of the normal plane at any point of a curve 

of double curvature - - - - - - -167 

130. To determine the equation of cylindrical surfaces in general - - 168 

131. General differential equation of cylindrical surfaces - 169 

132. The same determined otherwise - - - - - ib. 

133. Given the equation of the generatrix to determine the cylindrical surface 

which envelopes a given curve surface - - - - ib. 

134. If the enveloped surface be of the second order the curve of contact will 

be a plane curve and of the second order - 170 

135. To determine the general equation of conical surfaces - - ib. 

136. General differential equation of conical surfaces - 171 

137. The same determined otherwise - - - - - ib. 

138. Given the position of the vertex, to determine the equation of the conical 

surface that envelopes a given curve surface - - - - ib. 

139. Mongers proof that when the given curve surface is of the second order 

the curve of contact is a plane curve - - - - 172 

140. Shorter method of proof - - - - - - ib. 

141. Davies's proof that there is one point and only one from which as a ver- 

tex, if tangent cones be drawn to two surfaces of the second order, their 
planes of contact shall coincide - - - - - 173 

142. The plane through the curve of contact is always conjugate to the diame- 

ter through the vertex of the cone ----- 174 

143. Surfaces of revolution - - - - - - - ib. 

144. To determine the equation of surfaces of revolution in general - ib. 

145. Simplified form of the equation when the axis of revolution coincides with 

the axis of z -------- 175 

146. Remarkable case, in which the generatrix is a straight line - - ib. 

147. General differential equation of surfaces of revolution - 176 

148. A given curve surface revolves round a given axis, to determine the sur- 

face which touches and envelopes the moveable surface in every posi- 
tion - 177 

C 



XVlll CONTENTS. 

Article Page 

149. Example in the case of the spheroid ----- 178 

150. Characteristic property of developable surfaces - - - 179 
151. twisted surfaces - - - - ib^ 

152. Osculation of curve surfaces - - - - - - ib. 

153. At any point on a curve surface to find the radius of curvature of a nor- 

mal section -------- 18f 

154. Eider's theorem viz. at every point on a curve surface the sections of 

greatest and least curvature are always perpendicular to each other 182' 

155. Values of the radii of curvature of any perpendicular normal sections ib. 

156. Expressions for the radii of greatest and least curvatures - - ib. 

157. Peculiarities of the surface at the point where the principal radii have 

different signs ._._--- 183 

161. Means of determining when the signs are different - 184 

162. A paraboloid may always be found that shall have at its vertex the same 

curvature as any surface whatever at a given point - 185 

163. To determine the radius of curvature at any point in an oblique section. 

The theorem of Meusnier - - - *'' - - 186 

164. Lines of curvature ------- 187 

165. To determine the lines of curvature through any point on a curve sur- 

face --------- ib. 

166. Lines of curvature through any point are always perpendicular to each 

other 189 

167. On the developable surfaces, edges of regression, &c. generated by the 

normals to lines of curvature ----- 190 

168. Radii of spherical curvature ------ 191 

169. Given the coordinates of a point on a curve surface to determine the ra- 

dii of spherical curvature at that point - 192 

170. The radius of curvature of an oblique section any how situated with re- 

spect to the surface and to the axes of coordinates is now determinable ib. 

171. To determine the radii of curvature at any point in a paraboloid - 193 

172. Twisted surfaces - - - - - - - 194 

173. To determine the surfaces generated by a straight line moving parallel 

to a fixed plane and along two rectilinear directrices not situated in one 
plane -------- 195 

174. Two straight lines shown to pass through every point on the surface of 

a hyperbolic paraboloid ------ 196 

175. To determine the surface generated by the motion of a straight line along 

three others fixed in position, so that no two of them are in the same 
plane - 197 

176. To determine the surface generated when the directrices are not all pa- 

rallel to the same plane ------ 198 

177. Two straight lines shown to pass through every point on the surface of 

a hyperboloid of a single sheet - - - - - 199 

178. On twisted surfaces having but one curvilinear directrix - - 200 
i 79, To determine the general equation of conoidal surfaces - - ib. 



CONTENTS. XIX 

Article Page 

180. Equation of the right conoid - - - - - -201 

181. To find the equation of the inferior surface of a winding staircase - ib. 

182. To determine the differential equation of conoidal surfaces - -203 

183. The same determined otherwise - - - - - ib. 

184. Twisted surfaces having curvilinear directrices only - - - ib. 

185. To determine the general equation of surfaces generated by a straight 

line which moves along any two directrices whatever, and continues 
parallel to a fixed plane ___._. 204 

186. Determination of the differential equation of these surfaces - -205 

187. To determine the general equation of surfaces generated by the motion 

of a straight line along three curvilinear directrices - - - ib. 

188. Application of the preceding theory - - - ... 207 

189. Determination of the equations of the intersections of consecutive sur- 

faces - - - ... - - - 208 

190. Detei-mination of the general equations of developable surfaces - 210 

191. To determine the developable surface generated by the intersection of 

normal planes at every point in a curve of double curvature - - 211 

192. To determine the developable surface which touches and embraces two 

given curve surfaces - - - - - - ib. 

193. To determine the differential equation of developable surfaces in gene- 

ral 212 

194 The same determined otherwise - 213 

195. Envelopes, characteristics, and edges of regression - - - ib. 

196. The centre of a sphere of given radius moves along a given plane curve 

to determine the surface enveloping the sphere in eveiy position - 215 

197. On curves of double curvature ..... 216 

198. Expression for the differential of an arc of double curvature - - 217 

199. Osculation of curves of double curvature - - - - ib. 

200. Equation of the tangent deduced from the principles of osculation - 218 

201. To determine the osculating circle at any point in a curve of double cur- 

vature ----__.. 219 

202. General expression for the radius of absolute curvature - -221 

203. Conditions necessary for the circle to have contact of the first order only ib. 

204. Another method of determining the osculating plane - - - ib. 

205. Another method of determining the osculating circle ... 223 

206. Other and more simple expressions for the coordinates of the centre of 

the osculating circle - ■> - - - - - ib. 

207. To determine the centre and radius of spherical curvature at any point 

in a curve of double curvature - 224 

208. Determination of the equations of the edge of regression of the developa- 

ble surfaces generated by the intersections of consecutive normal 
planes to the curve - - - - _ _ _ 225 

209. To determine the points of inflexion in a eurve of double curvature - ib. 

210. On the evolutes of curves of double curvature ... -226 

211. Lines of poles ----.... 227 



XX CONTENTS. 

Article Page 

212. The locus of the poles the same as the locus of the characteristics - ib. 

213. Every curve has an infinite number of evolutes all situated on the de- 

velopable surfaces which is the locus of the poles - ib. 

214. Every curvilinear evolute of a plane curve is a helix described on the 

surface of the cylinder which is the locus of the poles of the plane curve 228 

215. The shortest distance between two points of an evolute is the arc of that 

evolute - - - - - - - - 229 

216. Given the equations of a curve of double curvature to determine those 

of any one of its evolutes - - - - - - ib. 

217. To prove that the locus of all the linear tangents at any point of a curve 

surface is necessarily a plane - - - - - 231 

218. Given the algebraical equation of a curve surface to determine whether 

or not the surface has a centre - 232 

219. To determine the equation of the diametral plane in a surface of the se- 

cond order which will be conjugate to a given system of parallel chords 234 

220. A straight line moves so that three given points in it constantly rest on 

the same three rectangular planes ; required the surface which is the 
locus of any other point in it - - - - - - 235 

221. To determine the line of greatest inclination - 236 

222. The six edges of any irregular tetraedron are opposed two by two, and 

the nearest distance of two opposite edges is called breadth; so that 
the tetraedron has three breadths and four heights. It is required to 
demonstrate that in every tetraedron the sum of the reciprocals of the 
squares of the breadths is equal to the sum of the reciprocals of the 

heights - 237 

Notes - 249—265 



ERRATA. 
Page 33, For article 20, read article 23. 

183, art. 157, at bottom of page, for — = r', read — = t § . 

184, articles 158, 159, 160, should not be numbered. 



THE 



DIFFERENTIAL CALCULUS 



SECTION I. 

ON THE 

DIFFERENTIATION OF FUNCTIONS IN GENERAL. 



CHAPTER X. 
EXPLANATION OF FIRST PRINCIPLES . 

Article (1.) All quantities which enter into calculation, may be 
divided into two principal classes, constant quantities and variable 
quantities ; the former class comprehending those which undergo no 
change of value, but remain the same throughout the investigation 
into which they enter ; while those quantities which have no fixed or 
determinate value, constitute the latter class. 

In algebra we usually employ the first letters, a, 6, c, &c. of the 
alphabet, to represent known quantities, and the latter letters, z, y, x, 
&c. as symbols of the unknown quantities ; but, in the higher calcu- 
lus, the early letters are adopted as the symbols of constant quantities, 
whether they be known or unknown, and the latter letters are used to 
represent variables. 

Any analytical expression composed of constants and variables, is 
said to be a function of the variables. Thus, if y "+ ax 2 + bx + c, 
then is y a function of x, because x enters into the expression for y ; 

1 



4, THE DIFFERENTIAL CALCULUS. 

y is also a function of x in the expressions y = a x + b, y — log. x 
+ ax 2 , &c. and, as in each of these cases the form of the function is 
exhibited, y is said to be an explicit function of x ; but, in such equa- 
tions as 

ax 2 -j- by 2 -f cxy + x~\- y, + e = 0, ;r s -f •-% — #?/ 2 = y J + ^ + c > & c - 

where the form of the function that ?/ is of ,r, can be ascertained only 
by solving the equation, y is an implicit function of x. 

Similar remarks apply to the equations 

z = ax 2 + by 2 + ex + e = 0, «c 2 + 6^ 2 + cxz + e = 0, &c. 
2 being an explicit function of x and y in the first, and an implicit 
function of the same variables in the second equation. 

If we wish to express that y is an explicit function of x, without 
writing the form of that function, we adopt the notation y = ~Fx, or 
y —fx, or y = cpx, &c. and, to denote an implicit function, we write 
~F(x,y) = 0,j[x,y) = 0, &c* 

(2.) Let us now examine the effect produced on the function y, by 
a change taking place in the variable a-, and, for a first example, let 
us take the equation y = mx 2 . Changing, then, x into x + /i, and 
representing the corresponding value of y by y f , we have 

y' = m (x + h) 2 
or, by developing the second number, 

y' = mx 2 -f- 2mxh -\- mh 2 . 

As a second example, let us take the equation y = x 3 , and putting 
as before y' for the value of the function, when x is changed into 
x + h, we have 

y > = (a? + /i) 3 = z 3 + 3^ 2 /i + 3xh 2 + /i 3 . 

We thus see, in these two examples, the effect produced on the 
function by changing the value of the variable, and, on account of this 
dependence of the value of the function upon that of the variable, the 
former, that is y, is called the dependent variable, and the latter, x, the 
independent variable. 

* In this general mode of expression, F, /, and cf>, are mere symbols, represent- 
ing the words a function of: thus, Fa:, or/a:, means a function of a:, the form of the 
latter differing; from that of the former. Ed. 



THE DIFFERENTIAL CALCULUS. 6 

Let us cow ascertain the difference of the values of each of the 
above functions of #, in the two states y and y'. In the first example, 

if — y = 2mxh + mh 2 . 
In the second, 

v > — y— 3^h + 3z/t 2 + /i 3 , 

so that, iii the equation y — mx 2 , if h be the increment of the variable 
x, we see that 2mxh + mh 2 will be the corresponding increment of 
the function y ; and, in the equation y = # 3 , if x take the increment 
h, the corresponding increment of the function will be Sx 2 h + 3xh? 
+ h 3 . 

We may, therefore, in each of these cases, readily find an expres- 
sion for the ratio of the increment of the function to that of the varia- 



ble, that is to say, the value of the fraction 
In the first case, 



y — v 
h ' 



ba the second, 



"i-^JL = 2mx + h. 



V = Zar + 3xk + h 2 . 



h 

It is here worthy of remark, that m both these expressions for the 
ratio, the first term is independent of h ; so that, however we alter the 
value of h. this first term will remain unchanged. If, therefore, h be 
supposed- to dimmish continually, and, at length, to become 0, the 
said first term will then express the value of the ratio. This first 
term, then, is the limit to which the ratio approaches as h diminishes, 
but which limit it cannot attain till h becomes absolutely 0. 

In the first of the foregoing examples, 2mx is the limit of the ratio 

t/ — y 

'1— — - ; or it is the value towards which this ratio continually ap- 
proaches when h is continually diminished, and to which it ultimately 
arrives when these continual diminutions bring it at length to h = 0. 
In the second example the limit is Sx 2 . 

(3.) We may now understand what is meant by the limit of the 
ratio of the increment of the function to that of the variable. It is the 
determination of this limit, in every possible form of the function, that 
is the principal object of the differential calculus. The limit itself is 



4 THE DIFFERENTIAL CALCULUS. 

called the differential coefficient, derived from the function ; so that, 
if the function be mar 2 , the differential coefficient, as we have seen 
above, is 2mx, and the differential coefficient derived from the func- 
tion, x 3 , is Sx 2 . 

In both these cases, as indeed in every other, the respective differ- 
ential coefficients are only so many particular values of the general 

y 7/ 

symbol £, to which ^—7— always reduces when h = 0. In the first 
Hi 

example above, £ = 2mx ; in the second, £ = 3x 2 . 

Instead of the general symbol £, a particular notation is employed 
to represent the limiting ratio, or differential coefficient, in each 
particular case ; thus, if?/ is the function, and x the independent va- 
riable, the differential coefficient is represented thus, -p. If z were 
the function, and y the independent variable, the differential coeffi- 
cient would be -7- ; the expressions -f- and -r- have, we see, the ad- 
dy dx ay 

vantage over the symbol £, of particularizing the function and the in- 
dependent variable under consideration, and this, it must be remem- 
bered, is all that distinguishes -~ or — from £ , for dy, dz, dx, are 

each absolutely 0. 

This notation being agreed upon, we have, when y = mx 2 , 

-^ = 2mx, 
dx 

and, when y = x 3 , 



dx 
As a third example, let the function y = a + 3x 2 be proposed, 
then, changing x into x + h and y into y\ we have 
y' = a + 3x 2 + 6xh + 3h 2 , 

.-. 2^2 = 6x + 3h, 

and, making h = 0, we have, for the differential coefficient, 

dx 
If in this example the function had been Zx 2 , instead of a + 3x , the 
differential coefficient would obviously have been the same. 



THE DIFFERENTIAL CALCULUS. O 

As a fourth example, let y = ax 2 ± 6, 

.-. y' == ax 2 ± 6 -f- 2aa?/i + a/i 2 , 

.♦. —7—^ = 2a.r + a/i .'.-j- = 2ax, 
li ax 

which would have been the same if the constant 6 had not entered 

the function. 

As a last example, take the function y — (a + hx) 2 or y — a 2 + 

2abx -}- 6 2 # 2 , which, when x is changed into x + /i, becomes 

2/' = a 2 + 2ab (x + h) -1- 6 2 (a? -f h) 2 
= ar + 2abx -f b 2 x 2 + 2 («6 -f b 2 x) h + 6 2 /i 2 , 

. y'—y 



h 



26 (a + bx) + 6 2 /i, 
^ = 2b (a + fa). 



It should be remarked, that of the two parts dy, dx, of which the 

dy 
symbol -~ consists, the former is called the differential of?/, and the 
cix 

latter the differential of .r. These differentials, although each = 0, 

have, nevertheless, as we have already seen, a determinate relation 

to each other ; thus, in the last example, this relation is such, that 

dy = 2b (a + bx) dx, and, although this is the same as saying that 

= 26 (a + bx) X ; yet, as we can always immediately obtain 

dii 
from this form the true value of £ or ~, we do not hesitate occa- 

dx 

sionally to make use of it. 

From the expression for the differential of a function, we readily 

see the propriety of calling -p a coefficient, being, indeed, the coeffi- 
cient of dx. 



CHAPTER II. 



DIFFERENTIATION OF FUNCTIONS OF ONE 
VARIABLE. 

(4.) Let/c represent any function of x whatever, then, if x be chan- 



6 THE DIFFERENTIAL CALCULUS. 

ged into x + h, the general form of the development of/ (x + /i), 
arranged according to the powers of h, will be 

f{x + h) =fx + AA + B/r + C/> 3 + D/t 4 + &c. 
as may be proved as follows : 

1. The first term of the development must befx. This is obvious, 
for this first term is what the whole development reduces to when 
h = 0, but we must in this case have the identity fx =fx; hence, 

fix is the first term. 

2. JYone of the exponents of h can be fractional. For if any expo- 
nent were fractional, the term into which it enters would be irrational, 
so that the development of/"(a? + h), containing an irrational term, 
f(x + h) itself would be irrational, since it is impossible that there 
should be an equality between an irrational expression and one that 
is rational. But, if /(a? + h) were irrational, fx would likewise be 
irrational, for the former function differs from the latter only in this, 
that x + h occupies in it the place of x in the latter ; and, therefore, 
as many values as fx has, in consequence of the radicals that may 
enter into it, so many values, and no more, must/(# -f- h) have. As, 
therefore, the first term of the development of/ (a: + h) has the same 
number of values asf(x + h) itself, none of the succeeding terms 
can contain a fractional power of h ; for, otherwise, the development 
would have more values than its equivalent function, which is absurd. 

3. JYone of the exponents of h can be negative in the general de- 
velopment. For, on the supposition that a negative power of h en- 
ters any term, that term would become infinite when h = ; but, 
when h = 0, the function is simply fx, which is not necessarily infi- 
nite ; so that, that development into which a negative power of It en- 
ters, cannot be the general development of f(x -f- h), understanding, 
by the general development, that which does not restrict x to any par- 
ticular value or values. By supposing a negative power of h to ap- 
pear in the development, we have just seen that such development 
would restrict x to the values determined, by the equation fx = go, 

or by the equation — = 0, to the exclusion of all other values, and 

is, therefore, not general.* 

* It has been shown, that the general development of jfo, when # becomes 
j; + h, is f(x -f h) =fx + Ah a -J- Bh b -\- Ch c -f- &c, in which the exponents 
a, b, c, &c. of the increment A of the variable, are whole and posilive numbers, it 



THE DIFFERENTIAL CALCULUS. 7 

As to the method of determining the coefficients A, B, C, &c. that 
will be investigated hereafter (in Chapter iv.) 

4. It must be here remarked, that, although the general develop- 
ment of every functiou of a; is of the above form, yet we are not to 
suppose that this form will remain unchanged whatever particular 
value we give to x, for particular values may be so chosen as to ren- 
der this form of development impossible, and such impossibility will 
be intimated by the assumed value of x rendering some of the coeffi- 
cients A, B, C, &c. infinite. The well known binomial theorem, 
which we already know to be of the above form, will afford an illus- 

yet remains to prove, that they will be represented by the series 1, 2, 3, &c. as as- 
sumed in the commencement of this chapter. 

It is already known, that the first term of the development of f(x -f- A), is the 
primitive function, fx, and that the remainder must disappear when h = 0, which 
remainder must then contain h as a factor, so that we shall have 

f(x + li)=fa + Fh (1), 

and thence 

p _ f(x + h)-fx ^ 
h 
P being a new function of a; and h, can likewise be developed, of which the first 
term will be the value P will assume when h = 0, which we will represent by p, 
and as the remaining terms must vanish when h = 0, they must contain h as a 
factor ; we thus have 

P = p + ah, 
which substituted in equation (1), gives 

f(x + h) = fx-\-ph + a/i 2 , &c (2) 

again we have 

in which Q, is another function of x and h, and may be developed the same as P ; 
the first term of tins development will be the value of & when h = 0, which we 
represent by q, and the remainder becoming zero when h — 0, must contain h as 
a factor ; we shall then have 

a = q + Rh, 
in which R is another function of a; and h. Expression (2), will thus become 

f(x -f h) —fx + ph + qh* + R/i3 + &c. 
being the general form of the development of f(x -j- h), in which p, q, r, £cc. are 
functions of a; alone, and correspond to the coefficients A, B, C, kc. in the article 
under consideration. Ed, 



8 THE DIFFERENTIAL CALCULUS. 



tration of this. Thus the general development offx = y/x -\- a, 
when we replace x by x -f /i, is, by the above mentioned theorem, 



f\(x + h) + a\ = V{x + a) + h = 

{x + a)% + J (x + a)~ ¥ & — | O + «)"" /i 2 + &c. 
where, in the case x = — a, all the coefficients become infinite, and 
the development, according to the positive integral powers of h, be- 
comes in this case impossible ; for the function then becomes merely 
sf h or /ii, in which the exponent of h is fractional. The impossibility 
of the proposed form of development in such particular case is always 
intimated, as in the example just adduced, by the circumstance 
of infinite coefficients entering it, for imaginary coefficients would 
imply merely that the function f(x + h) for the assumed value of # 
becomes imaginary, and not that the development failed. A particu- 
ar examination of the cases in which the general form of the deve- 
lopment fails to have place, will form the subject of a future chapter ; 
at present it is sufficient to apprise the student that such failing cases 
may exist. 

(5. ) By transposing the first term in the general development of 
f{x + /i), we have 

f{x + h) —fx = Ah + Bh 2 + Ch 3 + &c. 

.../(f^bzA^A + M + w + to 

h 
hence, when h = 0, 

#*'=A, 

dx 

from which result we learn, that the coefficient of the second term, in 
the development of the function f(x + /i), is the differential coefficient 
derived from the function fx ; so that the finding the differential coef- 
ficient from any proposed function, fx, reduces itself to the finding 
the coefficient of the second term in the general development of 
f(x-\- /i), or of the first term in the developed difference f(x + h) 

-fa. 

Having obtained this general result, we may now proceed to apply 
it to functions of different terms ; but it will be proper previously to 
observe, that those constants which are connected with the variable 
in the function/a?, only by way of addition or subtraction, cannot appear 
in the coefficient A ; because A, being multiplied by /i, can contain 



THE DIFFERENTIAL CALCULUS. 9 

no quantities which are not among those multiplied by x + h in 
f(x + h), or by x infx. 

(6.) To differentiate the product of two or more functions of the 
same variable. 

Let y, z, be functions of x, in the expression 
u — ayz. 
By changing x into x + h, the function y becomes 

y' =y 4- Afc + B/i 2 + C/i 3 + &c. . . . (1), 
and the function z becomes 

z' = z + A7i + B7i 2 + C7i°-f &c. . v . (2). 
Hence, when h = 0, we have from (1) 

y'—y = fy = A 

h dx 

and from (2) 

g ' — s = <*£ = A ,_ 
/& eta 

Multiplying the product of (1) and (2) by a, we have 

tt' = ayz -f- « (As -f A'«/) /i + &c* 

therefore, a | -p 2 + -j- ?/ ) being the coefficient of the second term 

of the development of u', we have 

du __ dy dz 

dx dx dx 

.\ du = azdy + aydz . . . (3). 
Hence, to differentiate the product of two functions of the same va- 
riable, we must multiply each by the differential of the other, and add 
the results. 

It will be easy now to express the differential of a product of three 
functions of the same variable. Let. 

u = wyz 
be the product of three functions of x ; then, putting v for ivy, the ex- 
pression is 

u = vz ; 
hence, by (3), 

* nf is that value which u attains when the functions y and z have varied by 
virtue of the variation h of the variable x on which they depend. Ed. 

2 



10 THE DIFFERENTIAL CALCULUS. 

du = zdv + vdz, 
butv = ivy ; therefore, by (3), dv = ydw -f wijd ; consequently, by 
substitution, 

du = 2?/dw + zwcfo/ + toz/cfe . . . (4), 
and it is plain that in this way the differential may be found, be the 
factors ever so many ; so that, generally, to differentiate a product of 
several functions of the same variable, we must multiply the differen- 
tial of each factor by the product of all the other factors, and add the 
results. 

If we suppose the factors to be all equal to each other, we shall 
obtain a rule to differentiate a positive integral power. Thus the 
differential of the function 

U = X m = X'X'X'X.... 

is 

du — x™- 1 dx + x m ~ l dx -f- x m ~ l dx ■+• &c. to m terms, 

that is 

i 7 du , 

du = maf 1- ax .*. — - = mx" 1- . 
dx 

This form of the differential is preserved whether m be integral or 
fractional, positive or negative ; but, to prove this, we must first dif- 
ferentiate a fraction. 

(7.) To differentiate a fraction. Letw = -, y and z being func- 
tions of x ; therefore uz — y, and duz = dy, that is, by the last article, 
zdu + udz = dy . \ du = — , 



or, substituting - for u, 



du = f%^?. 



z- 



Hence, to differentiate a fraction, the rule is this : From the product 
of the denominator, and differential of the numerator, subtract the 
product of the numerator, and differential of the denominator, and 
divide the remainder by the square of the denominator. 

(8.) To differentiate any power of a function. 

The form of the differential when the power is whole and positive 
has been already established. Let then 



THE DIFFERENTIAL CALCULUS, 11 



u = y 

m 

be proposed, y being a function of x, and^T being a positive fraction. 
Since u m = y m , 

.'. nu n ~ x du = my m ~ l dy, 
m y m ~ l _ in if 



•'' 


, du = 


n it" -1 J 


Now 








(m — 


mn — m 




l) n 


consequently, 




m 

, m 7T - ! 

du = — v 



dy. 



T 



n 



n 

Let now the exponent be negative, or 

u = y~ 



dy. 





.\u n — 


-m _ 

y m 




.'. du 1 


< = d-L 

y m 


but 






du n 


— nu n ~ l du, and d — = — 


y m ~ l j 
■ m -z&r d 




y m 


y 




.'. nu"~ l du = 


— - my~ m ~ : 



dy — — my~™~ 1 dy, 



and du = - — ■ — dy, 

nu n ~ l 

__tn 

or, substituting for u its equal y ^, we have 



m 



du = y dy. 

Hence, generally, to differentiate a power, we must multiply together 
these three factors, viz. the index of the power, the power itself dimi- 
nished by unity, and the differential of the root. 

This rule might have been deduced with less trouble, by availing 
ourselves of the binomial theorem, for, supposing in u = y p that the 
increment of the function y becomes h when the increment of x be- 
comes h, we have u' — (y + k) p and, by the binomial theorem, the 



12 THE DIFFERENTIAL CALCULUS. 

coefficient of the second term of the expansion of (y -f- k) p is pif^, 
whether p be positive or negative, whole or fractional. As, however, 
we propose to demonstrate the binomial theorem by means of the 
differential calculus, we have thought it necessary to establish the 
fundamental principles of differentiation, independently of this theo- 
rem. 

(9.) If it be required to differentiate an expression consisting of 
several functions of the same variable, combined by addition or sub- 
traction, it will be necessary merely to differentiate each separately, 
and to connect together the results by their respective signs. For 
let the expression be 

u = aw + by + cz -f- &c. 
in which w, y, z, are functions of x. Then, changing x into x -f- h 
and developing, 

w becomes w + Ah + B/r + &c. 

y y + All + Wh 2 + &c. 

z z + M'h + B'7i 2 + &c. 

.-. u u + (aA + 6 A' + cA" + &c) h + &c. 

.•. du = aAdx + bA'clx + cA"dx + &c. 

But 

Adx = dw, A'dx = cfy, A"dx = dz, &c. 
therefore 

du = adiv + fecfy + cJs + &c. 
that is, the differential of the sum of any number of functions is equal 
to the sum of their respective differentials. 

(10.) We shall now apply the foregoing general rules to some 
examples. 

EXAMPLES. 

1. Let it be required to differentiate the function 
y = 8x i — 3x 3 — 5x, 
By the rule for powers (8) the differential of 8# 4 is 8 X 4x 3 dx, and 
the differential of — 3a? 3 is — 3 X 3x?dx ; also the differential of 
— 5x is — 5dx ; hence (9), 

dy = 32x 3 dx — 9x 2 dx — 5dx, 

.-. $£ = 32* 3 — 9X 2 — 5. 
ax 



THE DIFFERENTIAL CALCULUS. 13 

2. Let y = (a? + a) (3a 2 + b). 
By the rule for differentiating a product (6), we have 

dy = (a? 3 + a) d (3x 2 + b) + (3^ 2 + b) d (x 3 + a), 
and (8), 

d (3x 2 + 6) = 6a?efo, d (a? 3 + a) = 3a^dar, 
.-. % = (a? 3 + a) 6xdx + {3x 2 + b) %x 2 dx, 

.-.-/ = 15a? 4 + Sa?b + 6ax. 
ax 

3. Let y = (ax + x 2 ) 2 . 

The differential of the root ax + x 2 of this power, is ad# + 2xdx, 

therefore, 

<% = 2 (ax + a; 2 ) (a + 2x) dx, 

...-1=2 (aa? + a? 2 ) (a+ 2a?). 
eta? 



4. Let y — V a-\-bx 2 . 
The differential of the root or function under the radical, is 2bxdx ; 
hence 

_i 5a? 

dy = l(a + bx 2 )- 2 2bxdx = da-, 

y/a + bx 2 



dy __ 6 



5. Let y = (a + 6a m ) n . 
The differential of the root or function within the parenthesis, is 
mbx m ~ l dx ; hence 

dy = n(a + bx 171 )"' 1 mbx m ~ l dx, 



-f = bmn {a + bx m ) n ~ 1 x m ~ l . 
ax 



6. Let y = 



(a + a; 3 ) 2 

The differential of the numerator of this fraction is 2xdx, and the 
differential of a + x 3 is Sx 2 dx, therefore the differential of the de- 
nominator is 2 (a + x 3 ) d^dx ; hence (7), 

_ (a + a? 3 ) 2 2a?eita — 6a? 4 (a + x 3 )dx _ 2aa? — 4a? 4 
- ^ (^T^) 4 " (a + a? 3 ) 3 dx ' 

dy _ 2a?(a — 2a? 3 ) 
' dx (a + a? 3 ) 3 



14 THE DIFFERENTIAL CALCULUS. 



7. Let y = \a+ V (b + ^\\ 



c c . - 



The differential of the root a + V(6 + — ) is £ (6 + — ) 

c c 2c 

d — , and d — — dx ; hence 

x 2 or* x* 



^ ^ 6 + ^ 



8. Let y = y/x 2 + \/« + x 2 . 

The differentia] of x 2 + \/ft + # 2 is 2xdx + (a + a^)~2 a7(^r r 

dy x x 

''' ~a~y ~~ — "i" z — = ~ ■ ■ _ 

V^+Va + x 2 2v/(a + ^)(^+\/a + ^) 

a; 

9. Lety — — . 

Va 2 + X 2 X ■ 



Multiplying numerator and denominator by Va 2 + x 2 -f a-, the 
expression becomes 

2^ ^ 

y = — + — ^ a 2 + a: 2 , 
° a 2 a 2 



\ % = d ^- + da:-f-^T d v/a 2 + a: 2 , 



dy_2x, sfa 2J r x 2 + ..,.** . 

~dx'~'a 2 ^ a 2 a 2 Va 2 + a? 2 

2a: a 2 + 2X 2 

— 7/T "r 



10. y = a 2 — x 2 .-.-/ = — 2x. 



a 2 Va 2 + ar 2 
eta 



11. */ = 4a; 3 — 2X 2 4- 7a: + 3 .-. -/ = 12a: 2 — 4a: + 7. 

dx 



THE DIFFERENTIAL CALCULUS. 15 

12. y=(fl4" bx) x 3 .-. ^ = Abx 3 + 3aj^. 

13. y = (a+ bx + cx* + k,c.) m .'. ~ = m(a + bx + ex 2 
+ &c.)^ 1 (6 + c:z + &c.) 

14. y = (a + bx 2 )K-.^ = ^¥~a~+bx^. 

"l* - 4-_±^_ %_ 6(1— a») 
10- y a_t "3 +Z 3 " c£r (3 + ^) 2 V^ 

, , ■ c dy b , c 

16. Sf = « + W*—.-.^= 57 j + ? -. 

17. i/ = (az 3 + 6) 3 + 27a 2 — x 2 (x — 6) .-. ^ = 6a* 2 

, 3J _m , 2(a 2 — 2^+ bx) 

(ax 3 + b) + = — '. 

Va 2 — x 2 



18. y 



dy _ 



X+Vl—X 2 " dx ' 71—^(1+23^/1-3') 

The functions in these examples are all algebraic, we shall now 
consider 

Transcendental Functions. 
(11.) Transcendental functions are those in which the variable 
enters in the form of an exponent, a logarithm, a sine, &c. Thus, 
a% a log. x, sin. x, &c. are transcendental functions : the first is an 
exponential function, the second a logarithmic function, and the third 
a circular function. 

To find the Differential of a Logarithm.* 

(12.) Let it be required to differentiate log. x. 

Put a for the base of the system of logarithms used, and let 

M = - t 

a — l — i — !) 2 + i — !) 3 — &c - 
then 

log. (1 + n) = M (» — £ n 2 + i ?i 3 — &c.) 
or, putting - for n, 

* Note (A'). 

f Algebra, Chap. vii. p. 219., or vol. i. p. 155, Lacroix's large work on the Dif- 
ferential Calculus. Ed. 



16 THE DIFFERENTIAL CALCULUS. 

x -f- h . , . . . . ■*„■ ,h h 2 . h* . % 

log.__ = log.(*+fc)_log.* = M(--^ + ^-&c.) 



/i M{ x 2x 2 ^3? &C ° 

This is the general expression for the ratio of the increment of the 
function to that of the variable. Hence, taking the limit of this ratio, 
we have 

d log, x _ M 
dx x 

If the logarithms employed be hyperbolic M — 1, and then 

dlQg.X __ 1 /pv 

do: x * *^ 

If they are not hyperbolic, write Log. instead of log. for distinction 
sake, then, since by putting a for 1 + win the series for log. (1 + ri), 
we have 

log. a = a_l_±(a_l)3+i(a_l)3_& c .=i 

it follows, from the expression (1), that 
d Log. x _ 1 
dx log. a . x 

Unless the contrary is expressed, the differential is always taken ac- 
cording to the hyperbolic system, because the expression is then 
simpler, log. a being = 1. 

From the preceding investigation we learn, that the differential of 
a logarithmic function is equal to the differential of the function di- 
vided by the function itself. 

(13.) To differentiate an exponential function. 

1. Let y = a? then log. y = x log. a .'. d log. y = dxlog. a, that 

is, — = dx log a .'. dy = y log. a.dx = log. a . a x dx. 

Hence, to differentiate an exponential, ive must multiply together the 
hyp. log. of the base, the exponential itself and the differential of the 
variable exponent. 

EXAMPLES. 



1. Let y — x (a 2 + x 2 ) s/ a 2 — x 2 .-. log. y = log. x -j- log. 
2 + a^+ilog. (a 2 — x 2 ), 



THE DIFFERENTIAL CALCULUS. i? 

dy _dx 2xdx xdx o? + ftV — 4a: 4 d x 

'** ~y ~ ~x a 2 + x 2 ~ a 2 — x 2 ~~ x (a 2 + x 2 ) (a 2 — x 2 ) 
therefore, substituting for y its value, we have, 
dy a 4 + a 2 # 2 — 4a: 4 
dx Va 2 — x 2 



\/ a + x + v a — a: 

2. t/ = log. — • 

V a -f- a: — v a — # 

Multiplying numerator and denominator by the denominator, the 

expression becomes 

2x — ■ 

^ = lo S* o 1~7 2 2 = lo S' * ~" lo S' C« — v'a 2 — ^ 2 ) 

2a — 2 v a — ar 



. % = c 1 _ g 

'da: >a: a ^ _ ^_ a 2 + a?2 > ar-Z^^^^o—V aW 2 ? 



a 



x \/a 2 



s/x 2 + 2aa: 

3 * 2/ = , , ■, , 2 ~ •"• lo S' ^ = * lo S- (* 2 + 2ax ) — i 
%/ar + xr — x 



log. (a: 3 + ^ 2 — a-), 

dy x + a , 3a: 2 + 2a: — 1 

• tL z^r dx —— dx — - 

* ?/ a: 2 + 2aa: 3 (a: :j + a: 2 — x) 

(1 — 3a) a: 2 — (a + 2) x — a 
Bx(x 2 + x—l) (a: + 2a) ' 

. 5% __ U 1 — 3 «) ^ — ( a + 2 ) ^ — « I J x 

dx sj (x 2 + x — 1)» (a: + 2a>* 

— d ij dx 

4. y — x" 1 ^- 1 .% log. y = m ^/ — 1 log. x .*. — = m V — 1 — , 

V x 

dy y . , . — 

.:-f-=m-V — l ~ m y/ — l . af* v -i-i. 
dx x 

From this example it appears, that the rule at (8) applies when the 
exponent is imaginary. 

5. y = a x *. 
In this example the variable exponent is x* ; hence, calling it z and 
taking the logarithms, we have 

3 



18 



THE DIFFERENTIAL CALCULUS* 



\og.z = #log. x .-. — = \og.xdx -f-^ ... dz = x*(l-{-\og.x)dx; 
z x 

hence, by the rule, 

dy x 

-j- = log. a . a* . x* (1 + log. x). 

6. y = e" ^ _1 + e~* ^~ l » where e is the base of the hyper- 
bolic system, 

.*. dy = e^" 11 " 1 ^ — 1 cfo — e -• ^~ l </ — 1 e£r, 

7. y = log. (log. a?).* Put 2 for log. x.'.y — log. ar.\ <% = — 

z 

bntdz = dlog.x = ^...dy = 

8. y = (log. a; n ) 



a? ^ log. a: dx x log. a? 

cfy _ mn (log. a; n ) m_1 



c/a? x 



9 - y = ^g. J (a + *) m («' + X T' («" + *)""} ••• !r = ~x- 



+ -77 



a' + x a" + a*' 
10. y = log. 



\/a — \/ x dx {a — x) V < 

11. y — c *»... ^ = e ** . af (i + i og . ap). 

12. y = (log.)"*.-.^ = - 



dx x log. # (log.) 2 x . . . (log.)" -1 a\ 



._ . n/ 1+a; 2 dy 1 

13. log. 7/ = ... -J. = _. __. 

14. y = a^, z being a function of x, .*. -^ = log.alog.6.a 62 6 2 .-7- 

fla? oa? 

* This means the logarithm of the logarithm of x, but the notation we shall 
hereafter adopt will be (log.) 2 *, and which we shall extend to circular functions ; 
thus, instead of sin. (sin. x), we shall write (sin.)%, the square of the sine being 
written without the parenthesis, thus, sin. z x. We may call such expressions as 
(log.)" x, (sin.)* x, &c. the nth log. of a:, the nth sine of #, &c. 



THE DIFFERENTIAL CALCULUS. 19 



15. y = of +a .-. J- = log. a . log. b . aT + * . h* 2 + *(2z+l) 



, cty log. « . « Iog * 

J da? a? 

]_7, y = e (Ios) * .*. — — = . 

" dx log. x (log.) 2 .r .... (log.)" -1 3C 

18. y = x** .-. -jt = ar^.af ^1 + log. a? (1 + log. 'a?}}. 

(14.) To differentiate circular functions. 

Let a? represent the versed sine of an arc of a circle whose radius 
is r, then r — x will represent the cosine of the same arc, and, by 
trigonometry, 

tan. _ r 
sin. r — x 
In this expression, x is the independent variable, and as this dimin- 
ishes, the arc itself diminishes, both vanishing simultaneously, and the 

i • • * an - • v • 

ultimate ratio of — — is - = 1 ; that is, the sine and tangent of an arc 
sin. r 6 

approximate to each other as the arc diminishes, and at length become 
equal. As the arc is between the sine and tangent when these be- 
come equal, the arc, also, must become equal to each ; therefore, we 
may conclude, that the ultimate ratios are as follows : 

tan. _ arc _ arc _ arc _ sin. tan. _ 

sin. ' sin. tan. chord chord ' chord 

1. Let it now be required to find the differential of sin. x. Chang- 
ing x into x -f h, we have (Gregory's Trig. p. 48) 

sin. (x + h) = sin. x + 2 sin. i h cos. (x + i h), 

sin. (x + h) — sin. x sin. \ h . , , . . 

,. -^ J = -tf- COS. (, + i h) ; 

■ sin. 4- h , d sin. x . 

when x = 0, f- = 1, .*. — = = cos. a?.*. asm. x = cos.xax. 

\h dx 

2. To differentiate cos. x. 

d cos. x = d sin. (■*• # — x) = — cos. (•! * — x) dx = — sin. a* cZo:. 

* The differentiation of circular functions may be obtained independently of 
these results. See the note (A) at the end of the volume. 



20 THE DIFFERENTIAL CALCULUS. 

Cor. As d cos. = — d ver. sin. .•. d ver. sin. x = sin. xdx. 
3. To differentiate tan. x. 

sin. x cos. a? d sin. a? — sin. x d cos. a? 



cZ tan. x = c£ 

cos. a? cos. ^a? 

that is 

_ cos. 2 a? + sin. 2 x 1 

ci tan. a- = ax = — aa? = sec. B x ax. 

cos, a? cos. j? 

4. To differentiate cot. a?. 

eZcot.a:=dtan. {\* — x) = — sec. 2 {\ it — x) dx = — cosec 2 #dfar. 

5. To differentiate sec. x. 

. , 1 sin. a? . 

a sec. x = a — = — dx = tan. x sec. x ax. 

cos. -a? cos. J x 



6. 


To 


differentiate 


cosec. 


x\ 






d 


cosec. x = d— — 
sin. 


X 




cos 


X 




sin. 


2 x 



dx = — cot. a? cosec. a; dx. 

These six forms the student should endeavour to preserve in his me- 
mory. 

EXAMPLES. 

1. y = sin. 2 x .-. dy = 2 sin. a? d sin. a? = 2 sin. x cos. a? da; 
= sin. 2x dx, 

dy . _ 

.*• j~ = sin. 2a?. 
da; 

2. y — sin. n x .-.dy — n sin. n_1 a? d sin. a? = n sin. n ~ 1 a? cos. xdx> 

. d y ■ _ 

• • j- = » sin. " ' a: cos. a?, 
aa? 

3. 2/ = cos. mx .•. dy = — sin. mxdmx = — m sin. mxdx, 

d V 
.*. -j~ = — mi sin. Mia:, 
aa? 

4. « = y tan. af, m beiftg a function of x, .•. du = tan. ar* cty-f- 
m a! tan. a", now d tan. a?' 1 = sec. z x n dx n = naf 1 sec. 2 x n dx, 

du dy 

•'• ~d~ ~ tan * ^ d - ~^~ v nxn ~ 1 sec * 2 ^ n * 

5. w = cot. a: y .*. du = — cosec. 2 x y dx?. Put z = a: v .*. log. 
s = w log. a?, 



THE DIFFERENTIAL CALCULUS. 21 

dz dx dx 

.'. — = y — + log. xdy .-. dz = dx v = {y — + log. x dy)x !/ 1 

and — = — cosec. 2 x v (- + log. a? -p) x*. 
dx x dx y 

6. y — xe cos - x .-. cfy — e c ° s ' * ^ + xe CM ' * <* cos - x ~ e cos ' * 
(1 — x sin. x) dx, 

.-. -i = e cos * (1 — sin. x). 
dx 

d (x e cos ' *) 

7. y = log. (a? e cos ' *) • '. dy = — — ^^-, and <2 (a? e cos - *) = 

e cos - * (1 — a? sin. x)dx, 

dy 1 — x sin. x 

dx x 

dy 

8. y = cos. x + sin. x \f — 1, .-. T7I— — sin.aH-cos.a?\/ — 1 

dy 

9. y = cos. x + cos. 2x + cos. 3a? + &c. .*. — = — (sin. x 

+ 2 sin. 2a? + 3 sin. 3a? + &c.) 

10. y = xe taxu *.:-^ = \l + a?sec. 2 a?J e^*. 

sin. w a? dy t sin. m_1 a?, . ,sin. m+1 x . 

11. w = .-. -r- = m I —\ + n\ — — \. 

° cos. n a? dx { cos. a?' t cos. ?i+1 a? > 

du .?/ , _ dy. 

12. u = sec. a? 5 ' ,\ — = tan. x v sec. a? y a^ - +lo£.a? -/*. 

aa? a? aa?' 

(15.) In the preceding trigonometrical expressions, the arc is con- 
sidered as the independent variable, and the lines sine, cosine, &c. 
as functions of it ; we shall now consider the inverse functions as 
they are called, that is, those in which the arc is considered as a func- 
tion of the sine, the cosine, &c. A particular notation has been pro- 
posed for inverse functions : thus, if y = Fa? be the direct function, 
then x = F -1 y is the inverse function, that is, if we represent the 
function that y is of x by y = Fa?, the function that x is of y will be 
denoted by x = F"~ l y. By thus representing these inverse functions, 
we may return immediately to the direct functions, considering, for the 
moment, F -1 in the light of a negative power of F, or an equivalent to 

■==- ; for then x = F" 1 y immediately leads to y = Fa.* Thus, if 
* See note (B'). 



22 THE DIFFERENTIAL CALCULUS. 

x = log. -1 ?/ •'. y = log. x, the inverse function log." 1 y meaning 
the number ivhose log. is y. In like manner, y = sin. -1 x means that 
y is the arc whose sine is x ; that is, returning to the direct function, 
sin. y = x. 

1. To differentiate y = sin. -1 x. 

Here the direct function is sin. y = x .•. d sin. y = dx, that is, 

dy 1 1 1 

cos. ?/c/?/ = ct# .*. -f- = = — — = — . 

dx cos.y VI — sin. 2 ?/ %/ 1 — x 2 

2. To differentiate y = cos. -1 37. 
cos. y = x .'. sin. ?/d?/ = dx .'. -J- — — - 



dx sm-y VI — cos. 2 ?/ 

3. To differentiate y = versin. -1 #. 

versin. 1/ = a? ••• sin. ydy = dx, 

. *i - _L_ - i_ 

* * dx sin.?/ "' ^2x — x 2 ' 

4. To differentiate ?/ = tan. -1 x. 

tan. y = x .\ sec. 2 ?/cty ^ dx .-. -~ = — == - — ■ -. 

J J J dx sec. 2 ?/ 1 + x 2 

5. To differentiate y = cot. -1 a?. 

cot. y = x .-. — cosec." ydy = dx.\— = 



dx cosec. 2 y 1 + x 

6. To differentiate y = sec. -1 x. 

d V 1 

sec. y = x, .'. tan. ?/ secy ay = dx .*. -j 1 = 



dx tan. ?/ sec. ?/ 

1 



x V x 2 — 1 
7. To differentiate y = cosec. -1 x, 

1 7 dy — 1 

cosec. y = x .-. — cotan. y sec. ydy = dx .•. ~ = 

J J J J cfo cot. ?/ sec. y 

1 



# V^ — 1 



EXAMPLES. 



dy 1 dw m 

1. y = sin. -1 ma: .♦. 7 = — — .*. -y-= — = 

dm* ^j __ ,^2^ d# ^! __ frftf 



THE DIFFERENTIAL CALCULUS. 23 

1 « dy , , . ■ , d sin." 1 x 2 

2. y - x sin. ar .\ ~- = sm. or + x . and 

ax ax 

d sin. 1 x 2 dx 



dx ^ i __ x i 

dy . 2X 2 

-* = sin. ar H == 

dx ji __ x 4 



3. «/ = cos. -1 a? -s/ 1 — a; 2 . Put x V 1 — a? 2 = z 

_ — & 

.•. ay = — — 

v/1 — z 2 



butcfe ?-(<J\— x 2 — — ===-)dx, and v/1 — z 3 = v /l_ a : 2 + ^ 
v/1 — ar 

m dy _ — 1 + 2a; 2 

" d# ^(i _a; 2 + a? 4 ) (1 —a?) 

x 

4. v = tan. -1 -. 

* 2 

% _ * . dy _ 8 

i + ( 



2^ t _i_ /• a \2 ' dx 4 + x 2 ' 

2~ } 

1 



5. y = cot. -1 (a + w^) 2 . •• dv = tt — ; n d (a + ma?V 

•* l+(a+ma?) 4 ' 

. . 2 . , - dw 2m (a + im) 

d (a + m^) 2 = 2 (a + ma?) mda 1 .%—- — — r ^ — ; > 

v ax l-\-(a-jrmxy 

6. vv = sec. -1 — .*. aw = — d — , and 

x m \ „ x m 

«V(^ - 1 

a _ am eta 



r.m+1 



dy mx m ~ l 

' dx ~ ~~ ,/ a 2 — ^m" 



__, V/1 + ^ 2 v/1 + * 2 . 

7. ?/ = cosec. i .•. dy = — a — 

J x J X 



VI + x 2 1 + a? i 



24 



THE DIFFERENTIAL CALCULUS. 



7 V 1 + x 2 . V 1 + a? . , V 1 + * 2 
x X s a; 

1 

eta, 



x 2 VI + x 2 

" * da; 1 + x 2 ' 
8., = (sin.-.^,.| = 2 si n.-,- 7r i_, 



9. 7/ = COS. 



% 



VI + z 2 ' ' dx 1 + x 2 ' 



\\ — x dy 
1. i/ = tan. -1 </- — ; .-. -£ 



1 + a; d* 2v/l— a^ 

11. i/ = (cot.- 1 *) 8 .*.-/ = — ^ , 2 cot.- 1 *. 

dx 1 -\"x* 

dy » 

12. i/ = sec. -1 *" >''-/-- 



dx xVx 2 ' 1 — 1 

cty 2 

13. y = cosec. -1 m* 2 .-. -—• = , , =< 
d* xVmx* — 1 



14. 2/ = versm.- 1 e r .-. -p- = 



d* n/2 — e* 

(16.) In the preceding expressions the radius of the arc is always 
represented by unity, but, as the differentials are frequently required 
to radius r, we shall terminate this chapter with the several formulas 
in (15) accommodated to this radius. We must observe, that as y 

and * are homogeneous in each of those forms, - is always a num- 
ber, so that this ratio in the limit, that is -—-, is a number. Hence, r 

dx 

must be introduced as a multiplier so as to render the numerator and 
denominator of each expression of the same dimensions. The for- 
mulas, therefore, become 

rdx 

d sin. -1 * = =■ 

Vr 2 — x 2 






w" 



THE DIFFERENTIAL CALCULUS, 25 

rdx 

d ccs." 1 % = — 



Vr 2 — ^' 
rdx 

V 2rx — x 2 

r?dx 
ci ran." 2 = 

d cot. -1 x = 



r* + x 2 
r*dx 



r^ + x 2 

r 2 dx 

d sec. -1 x — 



x\/ x 2 — r 

r^dx 
d cosec. -1 x = — 



aV^ 2 — r 2 

On successive Differentiation. 

(17.) Since the differential coefficient derived from any function 
of a variable may* contain that variable, this coefficient itself may be 
differentiated, and we thus derive a second differential coefficient. In 
like manner, by differentiating this second coefficient, if the variable 
still enters it, we obtain a third differential coefficient, and in this way 
we may continue the successive differentiation till we arrive at a co- 
efficient without the variable, when the process must terminate. 

Thus, taking the function y = ax\ we have, for the first differ- 
ential coefficient, ~ = 4ax 3 , as this coefficient contains x, we have, 
dx 

by differentiating it, the second differential coefficient = 12ax 2 ; 
continuing the process, we have 24ax for the third differential coeffi- 
cient, and 24a for the fourth, which being constant its differential 
coefficient is 0. 

If we were to express these several coefficients agreeably to the 
notation hitherto adopted, they would be 



first diff. coef. ~- = Aax 



dx 



dx 
second diff. coe£ — 5 — - = 12cm? 2 , 
dx 

* It must contain the variable, unless in the single case of its being constant. 

Ed. 
4 



26 THE DIFFERENTIAL CALCULUS, 

dx 
d- 



dx 
third difF. coef. -j = 24 ax, 

&c. 
But this mode of expressing the successive coefficients is obviously 
very inconvenient, and they are accordingly written in the following 
more commodious manner : 



first difF. coef. 

second difF. coef. 

third difF. coef. 

nth difF. coef. 



d V 
dx 

dx 2 
cPy_ 
dx" 
fry 



dxT ' 

in which notation it is to be observed, that d 2 , d 3 , &c. are not powers 
but symbols, standing in place of the words second differential, third 
differential, &c. The expresssions dx 2 , dx z , &c. are on the contrary 
powers, not, however, of x, but of dx : to distinguish the differential 
of a power from the power of differential, a dot is placed in the former 
case between d and the power. 

(18.) The following are a few illustrations of the process of suc- 
cessive differentiation : 

1. y = x m . 







dx 

dx 2 
d?y 
da? 


= mx™- 1 , 
= m (m — 

= m (m — 


1) iT-\ 
l)(ro- 


- 2) a:"*" 3 , 












d*y 

dx* 


= m (m — 


l)(m- 


-2)(m- 


■3) 


x*- 


1 






&c. 




&c. 










2. 


u = yz y 


both y 


and z being 


functions of x, 









THE DIFFERENTIAL CALCULUS, 



27 



du 

dx 

dru 



dz 



dy 



dx 
d-z 
~dx T ~ V ~dx T 



d 2 z 
~dx T 



dx 
dijdz 
dx 2 
dydz 



zd-y , dzdy 
dx 2 "*" dx* 
zd 2 y 



d 3 u 



dx" 


3 dx' 




&c. 


= log. x 




dx 


1 

X 


<Py_ 

da? 


1 

1? 


d?y _ 
da? 


2 

x 1 ' 



dx 2 
dyd 2 z 



dx 3 



f 3 



dx 2 

dzdhj 

~d^~ 

&c. 



+ 



*i 3 y 

dx 3 



2 -3 



&c. 



4. y = e*. 

dx 

d *y 

dx- 



d 'y . 

dx" ' ' x 4 
d 5 y 2.3-4 

eta 5 x b 

d 6 y _ 2-3-4-5 



^ 



dhj 
dx' 



&c. 



If instead of e the base were a, the several coefficients would be 
log. a • a*, log. 2 a • a T , log. 3 a . a*, log. 4 a • a T , &c. 

It appears, therefore, that exponental functions possess this property, 

, d"y . . 

viz. that ~- -f- y is always constant. 



5. y 



■ sin. x, 
dy_ __ 
dx 

d± 

dx 2 



COS. X 



= — sin. x 



d 3 y _ 
dx T ~ 

dx* 



COS. X 



&c. 



We need not multiply examples here, as the process of successive 
differentiation will be very frequently employed in the next two 
chapters. 



28 THE DIFFERENTIAL CALCULUS.- 



CHAPTER III. 

ON MACLAURIN'S THEOREM. 

(19.) If y represent a function of x, which it is possible to develop 
in a series of positive ascending powers of that variable, then will 
that development be 

where the brackets are intended to intimate that the functions which 
they enclose are to be taken in that particular state, arising from 
taking x = 0.* 

For, since by hypothesis 

y = A + Bx+ Cx 2 + Da? + Ea* + &c. .. (1) 



dy 

dx 


B + 2Cr + 3D* 2 + 4E^ 3 -f &c. 


d*y 

dx 2 


2C +2-3D^+ 3.4E:r 2 + &c. 


dhj 
dx 5 ~ 


2 . 3D +2.3. 4Ea? + &c. 


&c. 


&c. 


Let, now, x - 


= 0, then 
dy 



r d2 y -, _ „ c , . r _ i v d2 y -, 

A=2.3D,.D = -i-A 
W J ' OXJ ' 2-3 W J 

&c. &c. 



* This plan of enclosing the differential coefficient in brackets we shall usually 
adopt, when we wish to express not the general state of this function, but that 
state which arises from the variable taking a particular value. What that value 
is will generally be made known by the nature of the inquiry. 



THE DIFFERENTIAL CALCULUS. 29 

Hence, by substitution, equation (1) becomes 

which is Maclaurin's theorem for the development of a function, 
according to the ascending powers of the variable. We shall apply 
it to some examples. 

EXAMPLES. 

(20.) 1. Let it be required to develop (a + x)% the exponent n 
being any number whatever, either positive or negative, whole or 
fractional, rational or irrational. 

Put y = (a + x) n . . therefore . [?/] = a n 

•*• ~t^~ = n ( a + x y i ~ l [~^4 = nar ~ l 

(XX OjX 

% = n{n-l){a + xr* . . . [g-] = „(»- l )fl r» 

^L = n{ n-l ){n -2)(a + X )^, [g-j = 

n(n — 1) (» — 2)a n ~ 3 
&c. &c. 

Substituting these values for the coefficients in the foregoing theo- 
rem, there results 

(a + a?)" = a n + wa n l x + a"- 2 x 2 + — — L 

2 2*3. 

a n ~ 3 x 3 + &c. 

and thus the truth of the Binomial Theorem is established in its 

utmost generality. 

2. To develop log. (a + x). 

Put y = log. (a -f a?), therefore [t/] = log. a 

% _ 1 

dx a -\- x ' ' 

d?y _ 1 



dx 2 (a + x) 2 

dPy__ 2 

d* 3 (a + *) 3 
d 4 7/ _ 2-3 



dx' (a + #) 4 

&c. 



^-dx j 


= loo;. - 


dry 


1 
a 


d 3 y 


_ 2 


dhj 


2.3 




&c. 



30 THE DIFFERENTIAL CALCULUS. 

.-.log. (a + *) = log.a+*--^ + £ _£l + &c. 

dv 
Ify = log. x were proposed, then, since [?/], {-r-]* ^ c * are infinite, 

we infer, for reasons similar to those assigned at art. 4, that the de- 
velopment in the proposed form is impossible. 
3. To develop sin. x. 

y = sin. x .... [t/] = 

-g- = cos.* .... [.*] = .! 

_X = _ sin ., . ... ^=0 

cZ 3 t/ _d 3 ?/ _ 

d 'y r dH l l „ 

fi = cos.* . . . . [*t] = l 
dar 5 L dx° J 

&c. &c. 



1-2-3 1-2. 3-4-5 

4. To develop cos. #. 

t/ = cos. x .... [y] = 1 

£ = -*•• • • • -* = o 

^ = -cos., . . . -[^-]=-l 
&c. &c. 

... cos . x = , _ _^_ + t . 2 %, 4 - &c. 

5. To develop a r . 

y — a x therefore [y] = 1 

*7^ ■•..-•' [^1- a 

* A is put here for the hyperbolic logarithm of the base a, that is, for the ex- 
pression 

(a — 1) — £ (a — 1 )» -f J (o — l) 3 — &c. 



THE DIFFERENTIAL CALCULUS. 31 

d2y -av r-^h = a 2 

d^- Aa ' ' ' ' l d** j A 

d^- Aa ' • • * L ^ J_A 

&c. &c. 

, AV , AV , _ 
.•.*=1+A* + — + rJT5 + &c. 

which is the Exponential Theorem, 

Since A = log. a, we may give to the development the form 

a* — 1 + x log. a + - {x log. a) 2 + —— (x log. a) 3 + &c. 
2 2 . o 

For x — 1, we have the following expression for any number, a, in 
terms of its Napierian logarithm : 

1 1 

a — 1 + log. a + 2log. 2 a + ^— -^ log. 3 a + &c. 

changing a into the Napierian base, e, we have 

e *- i +x+ ^- + ^-f &c. 

which, when a: = 1, gives, for the base e, the value 

1 1 

e = 1 + X + g + ^-73 + &c. 

(21.) From the development of e x may be immediately derived 
several very curious and useful analytical formulas, and we shall avail 
ourselves of this opportunity to present the principal ones to the no- 
tice of the student. 



If, in the development of e*, we put zV — 1 for a-, we shall have 



zW — 1 



and, changing the sign of the radical, 



z 2 z 3 V — 1 



e -w-i = i _ zV _ ! ___ + ___ + _______ &c . 

If these expressions be first added and then subtracted, there will re- 
sult the following remarkable developments, viz. 



32 THE DIFFERENTIAL CALCULUS. 



C *VH1 


+ e^V-i 


c *v~i 


2 







1 1 • 2 + 1 • 2 • 3 • 4 &C * 



2^/ITT 1-2-3 ' 1 • 2 -3 -4 • 5 

Now it has been seen (examples 4 and 3) that these two series are 
also the respective developments of cos. z and sin. z\ hence, putting 
x instead of 2, we may conclude that 



,_*,/ - 1 



sin. x = = .... (1) 

2V — 1 

cos. x = = .... (2) 

where the sine and cosine of a real arc are expressed by imaginary 
exponentials. 

These expressions were first deduced by Euler, and are consider- 
ed by Lagrange as among the finest analytical discoveries of the age. 
(Calcul des Fonctions, page 114.) 

(22.) If for the real arc x we substitute the imaginary arc xs/ — 1 
/ x we shall have 



sm. {xV-1) = — =....(3) 
1 v — 1 



cos. (x sf — 1) = 2 .... (4) 

sin. 
Also, since — - = tan., it follows, from (1) and (2), that 
cos. 



«*v-i 


e~*v— 1 e w-i 


— 1 






— 1 tan. x — — 


— - — — 


* 






C W-1 + e -W~l e 2rV-l -J- ! 

By multiplying equation (1) by ± V — 1, and adding the result to (2), 
we have 

cos. x ± sin. x \/ — 1 = e*^- 1 . . . . (5 ;) 
or if we change x into mx, 

cos. mx ± sin. mx V — 1 = e ±^v-i .... (6), 
but e ± m **/- 1 is e ±w_1 raised to the mth power. Hence this singu- 
lar property, viz. 

* Multiplying the numerator and denominator of the second member of the 
equation by e*V-i. Ed. 



THE DIFFERENTIAL CALCULUS. 33 



(cos. x ± sin. x V — l) m = cos. mx ± sin. mx >/ — 1 .... (7), 

which was discovered by de Moivre, and is hence called De Moivre's 
formula. 

If the first side of this equation be developed by the binomial theo- 
rem, it becomes 

, to (to — 1 ) _ _ _ 

cos. m x db to cos. m ~ l xp -j- — cos. m ~ z xp* d= &c. 



p being put for the imaginary V — 1 sin. x. 

Now in any equation, the imaginaries on one side are equal to those 

on the other, (Algebra) ; hence, expunging from this expression 

all the imaginaries, that is, all the terms containing the odd powers of 

p, we have, in virtue of (7), 

to (to — I ) 
cos. mx = cos. m x — cos. m ~ 2 x sin. 2 # + 

to (to — 1) (to — 2) (to — 3 

~ t .3 t 4 cos. m ~ 4 x sin. 4 o? — &c. 



In like manner, equating sin. mx V — 1 with the imaginary part of 

the above development, and then dividing by y/ — 1, we have 

to (to — 1)(to — 2) 
sin.TO.r= mcos. m ~ l xsm.x — ZTTq cos. m " 3 j?sin. 3 a:+&c. 

From these two series the sine and cosine of a multiple arc may be de- 
termined from the sine and cosine of the arc itself. 

(20.) If in the formula (2) we represent e x ^'~ l by y, then e~ e ^-~ l 
= - ; therefore, 

y 

1 

2 cos. x = ii -\ — 

J y 

or, if in the same formula mx be put for x, we have 

1 

2 cos. mx = y m + — - 

3 ym 

and from these two equations we deduce the following, viz. 
if — 2ij cos. x + ! = . . . . (1) 
f™ _ 2y m cos. mx -f 1 =0 . . . . (2). 
Since these equations exist simultaneously, the latter must have 
two of its roots or values of y equal to the two roots of the former, 

5 



34 THE DIFFERENTIAL CALCULUS. 

and must, therefore, be divisible by it ; or, putting 6 for mx, we have 

y ^ _ 2y cos. 6 + 1 = (3), 

divisible by 

f — 21/ cos.— '+ 1=0... . (4). 

But cos. 6 = cos. (0 + 2n*), ?i being any whole number, and if — 
180° ; hence, making successively n — 0, = 1, = 2, &c, to w = 
m — 1, we have, since the first equation continues to be divisible by 
the second in these cases, 

& 

y2m — 2?/ m cos.d + l = (if — 2?/ cos. h 1) 

6 + 2« 
X(tf-2ycos.—^ r -+ I) 

6 + 4* 
X (^-22, cos.— — + 1) 

6 + 6* 
X (r/ 2 — 2i/cos. h 1) &c. torn factors. 

The truth of this equation is obvious, for, while the substitution of 

& + 2nif for 6 causes no alteration in the expression (3), the same 

substitution in (4) gives to that expression a new value, for every va- 

' , 6 6 + 2« 

lue of n. from n = to n = m — 1, for the arcs — , &c. are 

m m 

all different. As, therefore, the expression (3) is divisible by (4) 
under all these m changes of value, it is plain that these are its m 
quadratic factors. 

In this way may any trinomial of the form y 2m — 2hy m + 1 be de- 
composed into its quadratic factors, provided k does not exceed unity, 
for then k may always be replaced by the cosine of an arc. 

(24.) The geometrical interpretation of the foregoing equation, 
presents a curious property of the circle, first discovered by De 
Moivre. To exhibit this property, let P be any point 
either within or without the circle whose centre is 0, and 
let the circumference be divided into any number of 
equal parts, commencing at any point A, Join the points 
of division, A, B, C, &c. to P, then, since in the fore- 





THE DIFFERENTIAL CALCULUS. 35 

going analytical expression the radius OA is ex- 
pressed by unity, we shall have, by introducing 
the radius itself so as to render the terms homo- 
geneous, the following geometrical values of the 
above factors, where it is to be observed that 

Z POA = — and OP = y, 
m 

^2771 _ 2yf» cos. -j- 1 = OP 2 " 1 — 20P OT X OA OT cos. m ( AOP) + AQ Zm 

w a — 2« cos. ° f- 1 = OP 3 — 20P X OA cos. AOP + AO 2 = PA 2 * 

m 

f — 2y cos. g + 2?r + 1 = OP 3 — 20P X OA cos. BOP + BO 2 = PB 3 
in 

k + 1 = OP 2 — 20P X OA cos. COP + CO 2 = PC 2 



m 

&c. &c. &c. 

Hence, 

OP 2 ™ — 20F» X OA™ cos. m (AOP) -f OA 2 " 1 = PA 3 X PB 2 X PC 2 X &c. 
and this is Demoivre's property of the circle. 

(25.) If AOP = 0, that is, if P be upon the radius through one of 
the points of division A, then cos. m (AOP) — 1. Hence, 

OP 2 - _ 20P OT x OA 7 * + OA 2OT = PA 2 x PB 2 x PC 2 x &c. 
consequently, extracting the square root of each member, 

OP" 1 ^ OA OT = PA x PB x PC x &c. 
If the arcs AB, BC, &c. be bisected by A', B', &c. the circumfer- 
ence will be divided into 2m equal parts, and, by the equation just 
deduced 

OP 2 "* ^ OA 2m = PA x PA' x PB x PB' x &c. 
that is 

OP 2 ™ ^ OA 2ffl = (OP OT ^ OA™) PA' x PB' x &c. 
therefore, 

OP 2 ™ OA 2 "* 

Qp ; t " QAm = OP" + OA 771 = PA' x PB' x PC x &c. 

and these are Cotes's properties of the circle. 

(26.) If now we return to the expression (5), and suppose x = 

-, it becomes 

* Gregory's Trigonometry, p. 54, or Lacroix's Trigonometry. 



36 



THE DIFFERENTIAL CALCULUS. 



v /_l= e2 V- 1 ...log. v/_l = gV— 1, 
and 



From the second of these equations we get 
= 2l °gV — 1 __ |eg.(y — l) a _ log.- 

log. — 1. 
From the third 



* 1 tf 3 1 * 



( % /_ 1) v-. = 1 __ + T __. T ___._ + &c . 

Two very singular results ; first obtained by John Bernouilli. 
6. To develop tan. x. 

y = tan. x, therefore ... [#] = 

<% __ r rfM- 

-7— = sec.- .r r_ ■L.l — 1 

-^-2 sec.-* tan. * . . . [^3=0 

^ = 2 sec.'* (1 + 3 tan."*), [g] t* 2 

We thus see that the first two terms of the development are x + 
2x* 
I . 2 .3' but we sna M not continue the differentiation, since it does 

not make known the law of the series. The development will be 
more readily obtained by means of those already given for sin. x and 
cos. x, as follows : 

X -rT^S + l-2-3-4-5- &C - 
tan. x = 



x 2 a? 4 

&c. 



2 ' 1 • 2 • 3 -4 

therefore the development, found by actual division, is 

2x 3 ■ 16* 5 . 

tan. x = x h &c. 

1-2-3 1 • 2 • 3 • 4 -5 

but, to obtain the law of the terms and thus be enabled to continue 

them at pleasure, it will be best to apply the method of indeterminate 

coefficients. Assume, therefore, this fraction equal to the series 

Aj x + A 3 x 3 + A 5 x 5 + A, x~ + &c. 



THE DIFFERENTIAL CALCULUS. 



37 



Multiplying this by the denominator, we have this expression for the 
numerator, viz. 

x 3 



+ 



1-2-3 1.2-3-4-5 



&c = 



A! x + A 3 
A, 



1 -2 



x 3 + 



+ 



A 5 
A 3 
1 • 2 

A, 



x 5 + &c. 



1-2-3-4 

Hence, equating the coefficients of the like terms, there results 
A L = 1 therefore A x = 1 



A, 



A, 



1 



1 • 2 



1-2-3 



. . A 3 = 



_A L 1 

1 • 2 1-2-3 



+ 



A, 



1-2-3 

1 



5 1 • 2 ' 1-2-3 -4 1-2-3-4-5 

1 



-,A 5 = 



A-i 



1-21 -2-3-4 



+ 



1 • 2 • 3 • 4 • 5 
&c. &c. 

the law of these coefficients being such, that 

A 2n _i A2„_q , , Ai 



A 2 „+i 



1-2 1-2-3-4 



1 ' 2 . 3 . . . 2?i 



1 • 2 • 3 . . (2» + 1)' 
7. To develop tan. -1 x. 
y = tan. -1 x . . . . therefore .... [t/] = 
dy _ 1 



dx 1 + x 2 
d 2 y 2x 



dx 2 (1 + x 2 ) 2 

d 3 y _ 



dx' 
tfy 



2x 2-4^ 



(1 + x 2 ) 2 (1 + x 2 ) 3 



2 3 x 



+ 



2 A x 



2 4 • 3* 3 



dx' (1 + x 2 f (1 + x 2 ) 3 (1 + x 2 f 



&*-* 

& = • 



r d 3 y n 



A, 



38 THE DIFFERENTIAL CALCULUS. 

d 5 V _ 2* • 3 2».3V ff-3** r^ = o3 o 

d* 5 (1 + tf 2 ) 3 (1 + tf 2 ) 4 (1 + x 2 ) 5 ' ' l dx 5 J 
&c. &c. 

.*. y = tan. ?/ — i tan. 3 ?/ + j tan. 5 ?/ — -i tan. n y + &c. 
If y == 45°, then tan. y - 1 ; 

.-. arc 45° = 1 — i + i — | + &c. 
(27.) From this series an approximation may be made to the cir- 
cumference of a circle, but, from its very slow convergency, it is not 
eligible for this purpose. Eider has obtained from the above general 
development a series much more suitable, by help of the known for- 
mula, (Gregory's Trig., page 46,) 

. , , tan. a + tan. 6* 

tan. (a + 6)= T 

1 — tan. a tan. b 

for, when a + b = 45°, tan. (a + 6) = 1 ; therefore, 

tan. a + tan. 6 = 1 — tan. a tan. 6. 

If either tan. a or tan. b were given, the other would be determinable 

from this equation. Thus, if we suppose, 

1.1. 7 tan. b . n — 1 

tan. a = -, then - + tan. b = 1 , .♦. tan. b = — — — -. 

n n n n + 1 

Now the value of n is arbitrary, and our object is to assume it so 

that the sum of the series, expressing the arcs a, 6, in terms of their 

tangents, may be the most convergent. This value appears to be 

n = 2, or n = 3 ; therefore, taking n = 2, we have 

tan. a = i, tan. b = i, 

Hence, substituting in the general development a for y and -£ for tan. y, 

and then again b for ?/ and ^ for tan. 6, the sum of the resulting series 

will express the length of the arc a + b = 45°, that is 

1 1 , 1 1 , p 

arc . 45° = 2-3-^3 + 5-7^- Y7* + &C ' 

+ I i_+_J ^-+&c 

^2 3 . 3 3 ^ 5 . 3 5 7 . 3 7 ^ 

(28.) Another form of development, still more convergent than 

this, has been obtained by M. Bertrand from the formula 

2 tan. a 

tan. 2a == — 

1 — tan. z a 

* Lacroix Trigonometry. 



THE DIFFERENTIAL CALCULUS. 39 

For put tan. a = i, then tan. 2a = T 5 ^, therefore 2a Z 45°, because 
tan. 45° = 1 : from this value of 2a we deduce 

2 tan. 2a 120 

tan. 4a = — = 

1— tan. 2 2a 119 

.-. 4a 7 45°. 

Let now 4a = A, 45° = B, A — B = b = excess of 4a above 45°, 

then we have 45° = A — b. But 

,.._,. , tan. A — tan. B 1 

tan. (A — B ) = tan. b = — ■ =r = ^rr 

K J 1 + tan. A tan. B 239 

Consequently, if in the general development we replace y by a and 
tan. y by J, and then multiply by 4, we shall have the length of the 
arc 4a, and, since this arc exceeds 45° by the arc 6, if we subtract 
the development of this latter, which is given by substituting ^^ for 
tan. r/, the remainder will be the true development of 45°. Thus 

45° =4(i Ll + — — . __JL_+ & c .) 



( 1 &c.) 



i 1 — + ^- 

^239 3-239 3 5 -239 s 

This series is very convergent, and, by taking about 8 terms in 
the first row and 3 in the second, we find, for the length of the semi- 
circle, the following value, viz. 

* = 3 • 1415926535S9793. 
If we take but three terms of the first and only one of the second, we 
shall have it = 3 • 1416, the approximation usually employed in 
practice. 

(29.) The following examples are subjoined for the exercise of 
the student : 

8. To develop y = sin. -1 x. 

. sin. 3 y , 3 2 sin. 5 y . 3 2 • 5 2 sin. 7 y 

y = ■»■ y + jt^s + t^vjtt- 5 + T^w^rs^-r 

9. To develop y = cos. -1 x. 

, cos. 3 y 3 2 cos. 5 y . _ 

y = I ir — cos. y \- &c. 

3 2 * 1-2-3 1 2 -3 -4-5 ^ 

10. To develop y = cot. x by the method of indeterminate coef- 
ficients, as in example 6. 



*\ 



40 THE DIFFERENTIAL CALCULUS. 

1 x x 3 2x 5 

cot. x = & c 

a; 3 3 2 • 5 3 2 • 5 • 7 

11. To develop 7/ = (a + 6# -f. ex 2 -f &c.) B> 

(a + bx + ex 2 + &c.) n = 

: 3 +&C. 



i »_ii , w (n— 1) 



n(n— l)(n — 2) 

_| ___ a «_3 J3 

-|- na n — 1 cd 

This is the multinomial theorem of De Moivre. It is given in a 
very convenient practical form in my Treatise on Algebra. 



CHAPTER IV. 



ON TAYLOR'S THEOREM, AND ON THE DIFFER- 

ENTIATION AND DEVELOPMENT OF IMPLICIT 

FUNCTIONS. 

(30.) In the second chapter we established the form of the gene- 
ral development of the function F (x + h). We here propose to 
investigate Taylor's theorem, which is an expression exhibiting the 
actual development of the same function. The following lemma, 
must, however, be premised, viz. that if in any function of p + q one 
of the quantities p, q, is variable, and the other constant, we may de- 
termine the several differential coefficients, without inquiring which 
is the constant and which the variable, for these coefficients will be 
the same, whichever be variable. This principle is almost axiomatic. 
For as the function contains but one variable we may putp -f- q = x 
or F (p + q) = Fx, and whichever of the parts p, q, takes the in- 
crement A, the result F (.r + h) is necessarily the same ; hence the 
development of this function is the same on either hypothesis, and 
therefore the second term of that development, and hence also the 
differential coefficient. The first differential coefficient being the 
same, the succeeding must be the same ; therefore generally 
d n F (p + q) = d n F (p + q) 
dp n dq n 

whatever be the value of n. 



THE DIFFERENTIAL CALCULUS. 41 

Let now y = Fx, and Y = F (x + /i), and assume, agreeably to 
art. (4) 

Y = y + Ah + B/i 2 + C/i 3 + &c. 
A, B, C, &c. being unknown functions of x, which it is now required 
to determine. 

Suppose, first, h to be variable and x constant, then, differentiating 
on that supposition, we have 

~ = A + 2Bh + SCh 2 + &c. 
d/i 

Suppose, secondly, that x is variable and h constant, then the dif- 
ferential coefficient is 

dY dy . dA _ . dB dC 

~1~ = i + ~T~ h + -J- Jl + T- h + &c ' 
ax ax ax ax ax 

But by the lemma these two differential coefficients are identical, 
hence equating the coefficients of the like powers of h there results 

A = % B - — , C = — , &c. 

dx 2dx' 3dx' 



that is 



* _ d y r> _ d2 y l c- dhj 1 8lc 

A -^ B -dxJ'2' L -dx^'2T3 l8LC ' 



Hence the required development is 

_ . dy h . dhj h 2 , d?y h 3 * * , • 

tf h is negative, the signs of the alternate terms will be negative. 

When we wish for the development of the function Y = F(x-\-h) 
in any particular state, that is when x takes a given value, we have 
only to substitute this value for x in the general expressions for the 
coefficients previously determined, and we shall have the develop- 
ment according to the above form, that is, provided of course, that 
the development in such form is possible. But if the value chosen 
for x render the development impossible, the impossibility will be 
intimated to us from the circumstance of some of the terms becoming 
infinite,- as explained in art. (4). It may, however, be proper here 
to remark, that even in these cases of impossibility, the leading terms 
of the development as given by Taylor's theorem, are still true as far 
as the first term that becomes infinite. But as we propose to devote 

6 



42 THE DIFFERENTIAL CALCULUS. 

hereafter an entire chapter to the examination of the failing cases of 
Taylor's theorem, we shall not enter into the inquiry here. 

(31.)The theorem of Maclaurin may be easily deduced from that 
of Taylor, thus: 

Let x take the particular value x — 0, then 

'' , r dFx-h , d 2 ¥x. h 2 , r d?Fx- h? 
[Y] = F* = [F,] + l-^-l t + [-35^ + C^3 F¥T3 

+ &c. 
Now each of these coefficients is constant, and therefore indepen- 
dent of the value of h, hence h may take any value whatever, without 
affecting these coefficients; we may therefore call it x, it being 
observed that although x appears in the notation of the coefficients, 
it does not appear in the coefficients themselves. It follows, there- 
fore, that 

F* = [F*] + [-5-] * + c^r^ + [-3? ]f^ + &c - 

which is Maclaurin's theorem, before investigated. 

EXAMPLES. 

(32.) 1. To develop sin. (x + h) in a series of powers of the 

dy d 2 y . d?y 
arc/i. Leti, = sin.*.^=cos.*,^--sm.*-^ 

cos. a?, &c. hence, by Taylor's theorem, 

/i a ft 3 

sin. (a: + h) = sin. a? + cos. a? ft — sin. x j—^ — cos. x • 2 g 

+ &c. 
= si n.*(l_ — + — ^4- &c 

The series within the parentheses are respectively equal to cos. h 

and sin. h (p. 30,) hence the property 

sin. (a? + A) = sin. a? cos. h + sin. /fc cos. x. 

2. To develop cos. (x ~\- h). 

dy . d 2 n dhj . 

, = cos. x ••• J x = - bib. * .j£ = - cos. .r,^ = «m. « be 



THE DIFFERENTIAL CALCULUS. 43 

Hence 

h 2 b? 

cos. (x + h) == cos. x — sin. x h — cos. x + sin, x - — - — - 

1*2 \ ' A ' 6 

+ &c. 
^ cos . a7(1 ________ &c .) 

a h3 h5 « 

— sin. x ill •+■ — &c. 

v 1-2-3 1 •Z'3"1'5 

Hence the property* 

cos. (x + h) = cos. x cos. h — sin. x sin. h. 
From this property, and the analogous one for the sine of x + h 
deduced in last Example, the whole theory of trigonometry flows. 
By putting h = (m — 1) x, these two properties become 

sin. mx = sin. x cos. (mi — 1) x + sin. (m — 1) x cos. x. 
cos. mx = cos. x cos. (m — 1) x — sin. x sin (in — 1) x. 
Two equations which will be employed to abridge the expressions for 
the differential coefficients in the next Example. 
3. To develop tan. - 1 {x + h). 
y = tan. -1 o? 

~r- — r~ ~ cos « V- 

ax sec. y 

d 2 V n . dit . ^ 

—±-. = — 2 sin. y cos. y -~ = — sin. 2 y cos. 2 y. 

CLxr (Xx 

d 3 y dy 

-=~ = — 2 (cos. 2y cos. -y — sin. 2y sin. y cos. y)-f- 

(XX (XX 

dy 
= — cos. 3y cos. y -~ 

= — cos. 3y cos. 3 y 

d it dv 

~-^- = 2-3 (sin. 3y cos. hj -f- cos. By sin. y cos. hj) ~ 

(IX (XX 

dn 
= 2*3 sin. 4y cos. 2 y — = 2 • 3 sin. 4y cos. 4 y 

(XX 

* It should not be concealed from the student that the property here deduced is, 
in fact, involved in that Avhich we have employed to obtain the differential of a 
sine. If, however, we consider this differential deduced as in Note A at the end, 
then, the inference above, fairly establishes the property in question. 



44 



THE DIFFERENTIAL CALCULUS. 



d 5 y fa 

-^ — 2 • 3 • 4 (cos. 4y cos. 4 y — sin. 4y sin. y cos. 3 y) -~ 

= 2-3-4 cos. 5y cos. 2 y-j- = 2 • 3 • 4 cos. 5y cos. 5 i/ 

(XX 

&c. &c. 

hence 

* -1 / 1 7 X . 5 7 S i n - 2 V C0S « 2 V 1 

tan. (a? -f h) = y + cos. 2 y h 2 £ /i* — 

Ji 

cos. 3y cos. % fe3 sin. 4ycos. 4 y ^ 4 + cos. 5ycos. 5 y ^ __ ^ 
3 4 5 

4. To develop log. (a- -f h) according to the powers of h. 

log.(, + / ! )= 1 og.,+ M(J-£- + | r -| r ) +& c. 

5. To develop tan. (x + h) according to the powers of h. 

h 2 
tan. (x + h) = tan. # + sec. 2 x . h + 2 sec. 2 .z tan. x - — - + 

1 • 2 

h 3 
2 sec. 2 x (1 + 3 tan. 2 .r) + &c. 

1 . A • O 

(33.) By means of the theorem of Taylor may be obtained a very 
commodious and useful form for the representation and subsequent 
determination of the differential coefficient, when the function is 
complicated, thus : 

Let u = Fy, y being any function of x, which we may represent 

by y = Jx f and let it be required to find the expression for — . Let 

ax 

x take the increment /i, then since y = fx, the corresponding incre- 
ment of y will, by Taylor's theorem, be 

d y h + 5l Ji— 4- ^l /i3 _i_ *„ 

dx dx 2 1 • 2 ^ dx* 1 • 2 • 3 ~*~ C ' 

Call this increment h t then the corresponding increment of u will be 

that is, restoring the value of k 

+ &c. 



THE DIFFERENTIAL CALCULUS. 45 

Hence dividing by the increment h of the independent variable, 

and taking the limit as usual, we have 

dFy _du _ du dy _du dy 

dx dx dy' dx' ' dy dx 

du 
It appears, therefore, that the differential coefficient —is found by 

differentiating the function, on the hypothesis that y is the independ- 
ent variable, and then multiplying the coefficient thus obtained, by 
that derived from y considered as a function of x. 

(34.) The following examples will suffice to illustrate this mode 
of finding the differential coefficient. 
1. Let u = a? where y = 6*. 

1st. ~ = a y \og. a, 2nd. -- = b x log. b, 
dy & dx 5 



du _ du dy 
dx dy ' dx 
2. Let u = log. y, where y = log. x, 

du 1 dy 1 

dy y' dx x, 
du du dy _ 



log. a log. b. 



dx dy dx x log. x 



x 



3. Lettt = sin. ( ) 2 . 

a -f x 

# 2_ du _ d sin. y _ ^ dy _ 2ax 

V -_1_ ,,/ " ' ' rhi flit ' *'' A-r 



a-+ x } 5 " dy dy J ' dx (a + xf r 

du _ x 2ax 

' ' dx a -\- x } ' (a + xf 



4. Let u = cot. a y , y being = log. 



x 



Va 2 + x 2 

2 



du dy a 

— = — cosec. a y . a y log. a, ~° = 



cfa/ da? a?(a 2 + a? 2 ) 

dw «vi a2 

a y k>£. a . 



da? - t=> - • x ^ a 2 .j-^y 

(35.) Let us now take the more general function u = F (p, q) y 
p and q being functions of x, and suppose that when a? becomes a? + 
ft, p and q become p -\- k and g + h'. Call this latter q', then in 
consequence of the proposed change in the independent variable, the 
function will become F (g', p -f- &), in which it is to be observed q' 



46 THE DIFFERENTIAL CALCULUS. 

enters as if it were a constant, since it is unaffected by the increment 
k of the variable p. 

Hence by Taylor's theorem 

+ &c (1). 

But if in F (q r , p) we substitute for q' its equal q + k' we have 

du d 2 u l J2 

F(9>)=F(p,9 + *') = « + ^S'+^ I ^+&c..., ! (2) 

du 

,.' */*> = * + — j^ +& c....(3), 
op dp dp K 

and thus by continuing the differentiation may all the coefficients in 
(1) be developed according to the powers of ¥, but this first will be 
sufficient for our purpose. 

Substitute in the first two terms of (1) their developments (2), (3) 
and we have 

dif dil 

F (</ + &', p + k) = u + — k' + &c. + — k + &c. . . . (4) 

But k being the increment of the function p, arising from x taking 
the increment h, and k' being the increment which the function q takes 
from the same cause, it follows that 

dp . . d 2 p h 2 dq d 2 q k 2 

k = -fh + -jh- + &c, k =-lfc + — -?- - + &c. 

dx dor 1*2 dx dx z 1 • 2 

Hence by substitution in (4) we have finally 

, T . . c du dq , du dp,. . , a 

du _ du dq du dp 
' dx dq' dx dp ' dx' 
(36. ) Again : let there be three functions of x, viz. v = F (p, q, r) 
then when x becomes x -\- h let p, </, r become p + k, q -f k', r + 
k" respectively, and put r' for the latter, then in the function F (p -|- 
&, q + k\ r'), r' enters as a constant, hence as above 

where u == F (p, g, r') .♦. putting r + &" for r. 



THE DIFFERENTIAL CALCULUS. 47 

dv 

. dv ,,, . _ du dv dr „ 

„ = + _ fe: + & c. ) _ = ^ + Tq *+*°- 

dw 

^ = *L + ' dr fc" 4. &c. 

dp dp dp 

But h" = -j- A + &c. consequently 
^ , . , ,t, . 7„x , c -d» Hr , du dq dv dp 

+ &c. 
cZu _ du ^ r 1 c ^ ^9 1 ^ y ^JP 
"da? dr * dx dq ' dx dp' dx 9 
and so on for any number of functions. Hence the rule is to differ- 
entiate the expression with regard to each of its constituent functions 
severally, as, if all the others were constants, their sum will be the re- 
quired differential. 

Cor. lip is simply x then in the function u = F(x, q), 







du 
dx 


_ du 

dx 


du 


dq 
dx 






and 


in the function u 


= F 


{x,q, 


'•). 












du 
dx 


du 
dx 


du 

dq 


dq + 

' dx 


du 
dr 


dr 
dx 



(37.) -We must not confound here the— on the left, with that on 

the right, in these equations, for the former denotes the total differen- 
tial coefficient, of which the latter forms but a part, and is therefore 
called a partial differential coefficient. It is to be regretted, how- 
ever, that analysts are not agreed as to the best means of distinguish- 
ing total from partial differential coefficients, and accordingly in most 
works on the calculus the same symbol is applied indiscriminately to 
both ; a circumstance likely to prove a frequent source of perplexity 
to the learner ; and to avoid which we shall, throughout this volume, 
always distinguish the total differential coefficient by enclosing it in 
braces, so that the two equations above will be written thus : 
,du _ du du dq 
dx dx dq ' dx 



48 THE DIFFERENTIAL CALCULUS. 

.die. du du dq du dr 

dx dx dq ' dx dr ' dx' 

(38.) We shall now add a few examples, showing the application 
of the rule deduced in last article. 

EXAMPLES. 

- T . „ .du. du du du _ T 

1. Let u = cot. x y .-. i — l =_- + __. * Now 
ax ax dy ax 



du „ dx y 

-7- = — cosec/ x v —r- 
dx dx 



cosec. 2 x v — -— = — cosec. 2 x y yx y 



du dii _ dx y _ , du 

-7- . -r = — cosec." x y —— = — cosec. x y . x y log. x -^ 
dy dx dx ° dx 

2. Letw = 



V x 2 + (ar + V a? + $/ x + V *) 2 
Put x + n/ a; + 3/ a; + y a; = q .'. u = F (a?, 5), and 
efo« _ aqx 

dx {x 2 + q 2 Y 

du dq _ ax 2 dq dq _ 1 1 1 

^7 ' dx ~ ~ + g2) | * d? an ^~ 2V^ ^1 ^| ; 

hence 

eto _ ax* — aqx J_ _T J_. 

a (x 2 + g 2 ) 27 2a; 2 3* 3 4a: 4 

x 
3. Let w = log. tan. -, y being a function of x. 



. a: a? da: 
, d tan. - sec/ - . — , 

du dx _ 2/ _ y y - dx 


dx """ x 
tan. - 

a; 

7 dtan. - sec. 
au y 

dy V x~ 
tan. - 

y 


1 
tan. - w sin. - cos. - 

y y y 

x xdy 

y ~~tf xdl J 


tan. - if sin. - cos. 7 

y u y y 



hence 



IE DIFFEREN 


TIAL 


CALCULUS. 


s du } ~ 


y- 


dy 
dx 


W 


X 


X 



49 



y 3 sm. - cos. - 

J y y 



4. Let u = log. (.r — a -f \f x* — 2ax) 

5. Let w = (cos. a:) Bl " •* 

.du, , \ ■ , t sm - 2x n 

•*# j-T-t = (cos. x) Em * (cos. z log. cos. J ). 

e dx* cos. a? • 

Implicit Functions. 

(39.) Hitherto we have considered explicit functions only, or those 
whose forms are supposed to be given. We -shall now consider 
implicit functions, or those in which the relation between the indepen- 
dent variable x, and function y, is implied in an equation between the 
two, and which may be generally expressed by 
tt = F (ar, y) = 0. 

The deductions in article (36) will enable us very readily to 

find the coefficient — from such equations, without being under the 

OjX 

necessity of solving them, a thing indeed often impossible. 

If we turn to the corollary in the article just referred to, and sub- 
stitute y for q, we find 

e du, du , du dy 
\ — \ = — + — . — . 
*dx ax dy ax 

(XU 

But here u = F («r, y) = 0, therefore \ — \ =0, for u' — u being 



dx- 



. n u — u . 
always 0, — is always ; hence, 



du , du dii 

dx dy dx 
from which equation the differential coefficient is immediately deter- 
mina.ble : it is 

dy _ du, , du 

dx dx dy ' 
7 



50 THE DIFFERENTIAL CALCULUS. 

hence, having transposed the terms all to one side of the equation, we 
must differentiate the expression as if y were a constant, and then 
divide the resulting coefficient, taken with a contrary sign, by that 
derived from the same expression, on the supposition that x is a con- 
stant. 

EXAMPLES. 

1 . Let m = y 2 — 2mxy + x 2 — a = 0. 

du ^ du dii my — x 

.-. = 2my — 2x, — = 2y — - 2mx .'. -- = — . 

ax dy dx y — mx 

2. Let u = x- -j- Zoxy + y 3 = 0. 

du du n , « - dy x 2 -f ay 

.'. j- = _3x 2 — 3a»/,— = 3ax -f 3y 2 .-. -/ = —4,. 

ax "2/ cm: ax -+- 1/- 

If the second differential coefficient be required, we have 

dy dy 

dhj (ax + r) (2x + a ■£) + (x 2 + ay) (a + 2y ^) 

dx 2 (ax -f- ?/ 2 ) 2 

or substituting for ~ its value just found 
&y __ (ax + y 2 ) (2«x 2 + 2ry" — ax*— a*y) -\-(x* + ay) (a*x + ay 2 — Wy — 2oj/ 2 ) 

2xy* + 6axY + 2x 4 t/ — 2a 3 xy 

(aT+W 
2xy (y 3 + Saxy -f x 3 ) — 2a 3 xy 
(ax + i/) J ' 

that is, since x* + 3ax?/ -f y 3 = 0, 

dry 2a? xy 

dx 2 (ax -f- y 2 ) 3 ' 

3. Let mi/ 3 — xy = m to develop y, according to the ascending 
powers of x, 

du du n _ dw v 

dx dy J dx Smy 2 — x 

therefore, calling the successive differential coefficients p, o, r, &c. 

__ y — Sm tfp — xp 
9 (Smy 2 — x) 2 ' 



THE DIFFERENTIAL CALCULUS. 51 

3my*q -}- 2 • 3 myp 2 + xq (y — 3my 2 p — xp)(\2myp — 2) 
T (3mif — x) 3 (3my 2 — x)' 3 

&C. &LC. 

••• [«/] = 1. W = fcp M = 0, [r] =-^5. 
Therefore, by Maclaurin's theorem 

J 3m 3V 

4. Let ?/ 3 x — m 3 (y + *) = 0, to develop y according to the 
ascending powers of a?. 

Representing, as in last example, the successive differential- coeffi- 
cients by p, 9, r, &c. we have 

y 3 — m 3 _ _ 

9 - 3 *tf -- m 3 ' (!/ ^ } • ~(3xf - m 3 ) 2 * ' lql * 

■ M - 2 ' 3 ^ 2 - 3 m 3 (2tjp + xf) = 

' ' L J 3;n/ 2 — m 3 (3a^ 2 — w 3 ) 3 



.-. w = 



2 • 3 » 3 2-3.4 »?i ? 7> 2 2-3-4 



3.r?/ 2 — wi J (3^-z/ 2 — m 3 ) s 
&c. &c. 

Hence, by Maclaurin's theorem, 

__ _ ** 3a?1 

? x — ^T — 

5. Let y 2 + 2ry + a: 2 = a 3 , to find -.— 

.-. ^ = — 1. 



6 . Let-^— = 2y ^fL=L* - , + 6 to find S 

(ar — a) J "* x — a dx 

dy _3x — 26 — a 
" dx~ 2^/x — 6 ' 

7. Let ?/ 3 — 3y -f a: = 0, to develop ?/ according to the as- 
cending powers of x, 



52 TlflB I>IFTERENTIAL CALCULUS. 

8. Lei vnxf'x — y = tn, to develop ?/ according to the ascending 
powers of x. 

y = — m — m A x — Sni'x 2 — &c. 

9. Let my* — x"y = raf, to develop y in a series of descending* 
powers of x. 

ii = — m — &c. 



CHAPTBH V. 
ON VANISHING FRACTIONS. 

(40.) It is here proposed to determine the value of a fraction 



F 



x 



fa 

in the case in which, by giving a particular value a to the variable, 
both numerator arid denominator vanish, the fraction then becoming 

Fa _ 
~~fa~ ~" 0' 
As such a form can arise only from the circumstance of the same 
factor x — a being common to both numerator and denominator, it 
is plain that if we can by any means eliminate this factor before our 
substitution of a for x, we shall then obtain the true value of the fraction. 
Sometimes the vanishing factor is manifest at sight, and may be 
immediately expunged, as, for instance, in the fractions 
(x — a) x x 2 — a 2 a 2 — lax -f- x 2 



bx — ab* (a — x)' J 



, &c. 




each of which becomes- when x — a, and obviously contains 

the factor x — a in both numerator and denominator. In these cases, 



* This will be effected by substituting - for x 3 , which will transform the equa- 
tion into my*z — y = m, then developing y according to the ascending powers of 
z> and afterwards restoring the value of x. 



THE DIFFERENTIAL CALCULUS. 53 

therefore, we at once see that the values of the fractions when x = a 
are, severally, 

T , GO , 0, &C. 



In certain other cases the value, although not so easily seen as in 
the foregoing instances, may, nevertheless, be soon ascertained, by 
performing a few obvious transformations on the proposed fractions. 
Take the following example : 

\/ x — >/ a -\- V (x — a) 
Vx 2 —a* 
which becomes £ when x = a. 
This fraction is the same as 

s/x—Ja 1 



\/ x 2 — a 2 \f x + a 

and the first of these terms is the same as 



x — a s/ x — a 



^ ( x *-. a *)(Vx+ Vfl) 2 >/ (x + a) W x+ V a) 

and this when x = a is = 0, therefore the value of the proposed 

1 

fraction when x = a is — =• 
s/2a 

(41.) But the most direct and general method of proceeding de- 
pends upon the differential calculus, and upon the development of 
functions, and the principal object of the present chapter is to ex- 
plain this. 

We shall premise the following lemma, viz. In the general de- 
velopment 

„.', „ . dFx . , d 2 Fx h 2 , „ 
F(x + h) = Fx + -— h + — - —— -f &c. 
ax dor 1 • 2 

it is impossible that any particular value given to x can cause Far, 
and at the same time all the differential coefficients, to vanish. 

For if such could be the case, then, for that particular value a, we 
should have 

F (a + h) = 0, 
whatever be the value of h ; but Fa = 0, and therefore the prece- 



54 THE DIFFERENTIAL CALCULUS. 

ding equation can exist only when h = 0, whereas the hypothesis 
supposes it to exist independently of the value of h. 

Let now, in the proposed fraction, * be changed into x + K then, 
by developing both numerator and denominator, it becomes 
Fx + F'xh + F"xh 2 + F'"xh? + &c. 
fx +f'xh +f"xh a + f" , xti i + &c. ' " * ' (1 ^' 
where F'x, F''x, Sac, f'x, f'x. &c. are put for the successive differ- 
ential coefficients divided by as many of the factors 1 • 2 • 3, &c. as 
there are accents. 

If in this we substitute a for x, then, since both F:r and/x vanish, 
the fraction becomes, after dividing numerator and denominator by 
h, 

F'a + F"ah + F" W + &c. 
f'a +f"ah +f'"ah 3 + &c ' * * ' (2) ' 

and this fraction when h = must obviously be equal to , that is 

fo 
Fa _ F'a 

If, however, both F'a and f'a are also 0, then, expunging these 
terms from the fraction (2) and dividing numerator and denominator 
again by h, we have, when h = 0, 

' Fa - F " a 

Fa 
and so on, till we at length obtain for -r- a fraction of which the nu- 

J a 
merator and denominator do not both vanish, and such a fraction we 
eventually shall obtain by virtue of the preceding lemma. 

Hence the following rule to determine the value of a fraction whose 
numerator and denominator both vanish when x = a, viz. For the 
numerator and denominator substitute their first differential coefficients, 
their second differential coefficients, and so on till ive obtain a fraction 
in which numerator and denominator do not both vanish, for x = a, 
this will be the true value of the vanishing fraction. 

EXAMPLES. 

1. Required the value of when x = 0. 



THE DIFFERENTIAL CALCULUS. 55 

— = log. a . a* — log. b ,b x .-. — — = log. a — log. b = log. y 
J J 

x * Sx -\- 2 

2. Required the value of — „ , n when a? — 1 . 

ar — 6ar + Sx — 3 

F'x _ 3a? 2 — 3 F'a _ o 

~fx~ ~ 4a? 3 — \2x + 8 '*' Ya ~ ° ' ' 
differentiating again 

F" F'a: 6x m F"a _ , _ 

f ~fx~ 122^—12 ''' /"„ ~~ ° 

3. Required the value of , when x = 90°. 

sin. a: + cos. x — 1 

F'a? _ cos. a? + sin. x F'a _ 

f'x cos. x — sin. x ' ' f'a 

4. Required the value of 

x + . r 2 _ ( n -L. i)2 ^+i _j_ p n2 _|_ 2/z — 1) £ n+2 — nV+* 

when a? = 1. 

F'a?_ 

/*" 

1 + 2a? — (n + l) 3 x n + (n-{-2)(2n 2 -\-2n-- l)x n+1 — n 2 (n + S)x" +2 
3(l-a?7 
F'a _ 

■'" A ~ 0e 
F"a?_ 
/"a? ~ 

2— n(?i+l)V- 1 +(n+l)(2?i 3 +6?i 2 +3n— 2)a: n — (7i+2){n 3 -\-3?i 2 )x'+ i 

______ 

F'a __ 
'''J 7 a~0 

F'"a? 



■(n 8 - K)(n+l)V- 5 -j-(n 2 +n)(2n 3 H-6)i 2 + 3n-2)£"- 1 (n+i)( n +2)(n 3 H-3fl 2 )x" 

__ 

F'"a __ » (» + 1) (2n +1) _ Fa 
•'* 7^ 6 ; fiT 



56 



THE DIFFERENTIAL CALCULUS. 



5. Required the value of - when x = 0. 

XT 

F"a __ 1 
J 17 a~'¥a 

x sin. x 90° 

6. Required the value of when x = 90°. 

cos. x 

F'a 

1 # 

7. Required the value of when x = 1. 

cot. a? - 
2 

F'g_2 

/'« * 

8. Required the value of = ; when x — a. 

log. a — log. x 

F'a 

f a 

(42.) If, in the application of the foregoing rule, we happen to ar- 
rive at a differential coefficient, which becomes infinite for the pro- 
posed value x = a, we must conclude that the development accord- 
ing to Taylor's Theorem is impossible for that particular value of the 
variable ; and that, therefore, the rule which is founded on the possi- 
bility of this development becomes inapplicable. The process, how- 
ever, to be adopted in such cases is still analogous to that above, 
depending upon the development of the numerator and denominator 
of the proposed fraction ; but here this development must be sought 
for by the common algebraical methods. 

Fx 
As before, let ~~ De a fraction which becomes £ when we change 
J x 
x into a. Substitute a -f- h for .r, and let the terms of the fraction be 
developed according to the increasing powers of h, either by involu- 
tion, the extraction of roots, or some other algebraical process, then 
we shall have 

F {a + h) kh a -f Bh f3 + &c. 



/(« + h) A^' + B'^' + fcc. 



THE DIFFERENTIAL CALCULUS. 



57 



a and a being the smallest exponents in each series, (3 and fi' the 
next in magnitude, and so on. Now these three cases present them- 
selves, viz. 

1°. a > a' ; 2°. a = a' ; 3 D . a < a'. 

In the first case by dividing the two terms of the fraction by h a ' t 
and then supposing h = 0, there results 
Fa 

7^=T = °- 

In the second case the result of the same process is 
Fa _ A 

>~ "" A"'" 
In the third case, by dividing the two terms of the fraction by /&*, 
and then supposing h = 0, the result is 

Fa A 

7JT = -o =«• 

It appears from these results that the development of the numera- 
tor and denominator need not be carried beyond the first term, or that 
involving the lowest exponent of h,* and according as the exponent 
in the numerator is greater than, equal to, or less than that in the de- 
nominator, will the true value of the fraction be 0, finite, or infinite. 
We have, therefore, the following rule : 

Substitute a -f h for x, in the proposed fraction. Find the term 
containing the lowest exponent of h, in the development of the nu- 
merator, and that containing the lowest exponent of h in the develop- 
ment of the denominator. If the former exponent be greater than 
this latter, the true value of the fraction will be 0, if less, it will be in- 
finite. But if these exponents are equal, divide the coefficient of the 
term in the numerator by the coefficient of that in the denominator, 
and the true result will be obtained. 

This method, which is applicable in all cases, may frequently be 
employed advantageously, even where the preceding rule applies. 

EXAMPLES. 

(x* — Zax 4- 2a 2 ) ^ 

9. Required the value of ~ when £ == a. 

(*» — a?)? 

* The first term which actually appears in the development is of course meant 
here. Those which may vanish in consequence of the coefficient vanishing not 
being considered, 

8 



58 THE DIFFERENTIAL CALCULUS 

Substituting a + h for .r, we have 

F (a -f h) _ ffi (h — ay (— ah) % + &c. 

/ (« + A) fc J ^ a2 + 3ah + fca ^ (3a 8 fc)s + &c 

Since the exponent of h in the numerator exceeds that in the de- 
nominator, we have 

F« ■ 

-7T- =0. 

fa 



10. Required the value of II — . — — — . when x 

V jr — cr 
■ a (see p. 53.) 
Substituting a + h for .r. 

F (« + /*) (a + *).* — «* + ** &* + &c. 



/ (« + fe; fc i ( 2 a + /i)i (2ah)i -f &c. 

Fa _ I_ 

11. Required the value of— ■ — — - when x = a* 

(1 + x — a)° — 1 

Substituting a + h for .r. 

F (a -h /() ffi (2a + ft)'* -f h h + &c. 

/ (a + A) = "~Tl'~-r- A)" — 1 = 3ft + &c. 
Fa _ i 

fa 

1 o r» • j ,u , r G ( 4fl3 + 4 ^) * — «^ ~ a ' J u 

12. Required the value of when a 

(2a 2 + 2xr)^ — a — x 
a. 
Substituting a -f- ft for x. 

F (a + ft) a (8a 3 + 12a 2 ft + 12aft 2 + 4ft 3 )^ — 2a 2 — ah 



f(a + h) 



(4a 2 + 4aft + 2ft 2 )^" — 2a — ft 



* To develop this according to the ascending powers of h we must write it 
thus: ( — a -f- ^) an d apply the binomial theorem when we have the series 
( _ a )§ _j_ f a _J ft -f &c. 



THE DIFFERENTIAL CALCULUS. 5U 



which, by actually extracting the roots indicated, 

h 2 

a(2a + h+ ^ + &c. — 2a — h) 



l 2 

2a + h + — + &c. 

4a 

«(^ + &c.) ^ Fa 


__ 2a — A 

1 

"2" 

.. = 2a. 


A 2 fa 
4a 


1 

4a 



This example is perhaps more easily performed by differentiation, 
according to the first rule : thus 



F'a; 


a (4a 3 + 4r 5 ) » 4.r 2 — a F'a 


A" 


(2a 2 + 2x 2 ) -1 2^7 — 1 ' * /'a ~~ 


F"a 


— a (4a 3 + Ax 2 ) s 32^ -f « (4a 3 + 4a ; ') » 8x 


/>" 


-3 1 ■ 
— (2a 2 + 2x 2 ) 2 4.r + 2 (2a 2 -f 2a; 2 ) » 




F"a 2 



a 

(43.) Having thus seen how to determine the value of any fraction 
of which the numerator and denominator become each for particu- 
lar values of the variable, we readily perceive how the value may be 
found when particular substitutions make the numerator and denomi- 
nator each infinite. For if — rr- = — then obviously 

fa <*> 

1 
Fa _ ~fa~ _ 

jr^jr - 

Fa 

So that if we find, by the preceding methods, the value of this last 
fraction, the value of the proposed fraction will be also obtained. 
The following example will illustrate this. 

1 x 
tan. (- * . -) 
v 2 a' 
13. Required the value of 3 _, .-$ &— \ when x = a. 



60 iUE DIFFERENTIAL CALCULUS. 

la this case the fraction takes the form .2L, therefore. 

go 

1 
•fl. _ Q (■ r? — g = )*~ 3 . F'j: _ 2a 1 .r- 3 

F cot. (A cr . -) ^ — cosec. J (A * . -) — 

F'a 2 4a 

/'a tf # 

~ " 2a 
(44.) By the same principles we may also find the true value of a 
product consisting of two factors, which for a particular value of the 
variable becomes the one and the other cc. For if Fa = and 
fa ~ co , then, 

F«x/, = I?=? 
I J 1 

We shall give an example of this. 

14. Required the value of the product (1 — x) tan. (J- *x) when 
x === 1. 

In this case the first factor becomes and the second co . 

If = 1 — x = 1 —J? _ „, x „ = 1_ 

1 1 cot. (A «rar) X -* * ' „ " * 

T - t->-t ' 2 cosec. 3 ***)7 

/* tan. (l^) 

1 2 

.-. F'a Xf'a ==—• = -. 

(45. ) And finally, by the same principles, the true value of the dif- 
ference of two functions may be ascertained in the case where the 
substitution of a particular value for the variable causes each of them 
to become infinite. 

For if Fa = co and fa = co , then 

1 1 

>~F^__0 

Fa ~> 1 -Q 

Fa x fa 
The following example belongs to this case : 






THE DIFFERENTIAL CALCULUS. 61 

15. Required the true value of the difference * tan. x — \ic 
sec. x, when x — 90°. 

1 1 1 _ __1 

fx Fx 4- < sec. x x tan. x _ xsin.x — £<r 



1 1 



Fx X fx x tan. x X ^ # sec. x 

by substituting for sec. a-, and then dividing numerator and 

denominator by 

i # x tan. x 

x cos. x -f sin. x 
. •• X x — f x = : . \ 1 a — fa = — 1 . 

^ — sin. a; J 

It should be remarked that in this, as well indeed as in the prece- 
ding cases, the transformation requisite to reduce the expression to 
the form £ in many instances at once presents itself to the mind, 
when of course it will be necessary to recur to the preceding formu- 
las. The example just given is one of these instances, for since 

sin. x 1 

tan. x = , and sec. x = , the proposed expression at 

cos. x cos. x 

once reduces to 

x sin. x — 4- it 



cos. X 
which is the required form. 

(46.) We shall terminate this chapter with a few miscellaneous 
examples for the exercise of the student. 

x n l 

16. Required the value of , when x = 1. 

x — 1 

Ans. n. 

, _, ~ . . , _ _ ax 2 + ac 2 — 2acx . 

17. Required the value of — — -. ; — j—^ when x = c. 

ox — 2bcx -f- 6<r 

Jlns. T . 
b 

x 3 — ax 2 — a?x -j- a 3 

18. Required the value of , when x = a. 

x 2 — a 2 

Am. 0. 

* This is obtained by multiplying the last fraction above and below by x tan. 

, . . sin. x . 1 
x X \* sec. x, then writing for tan. r, and for sec. x. Ed. 



<>^ THE DIFFERENTIAL CALCULUS. 



(2a 3 x — x v ) 2 — a (a?x)* 
value of 1 i L. 



19. Required the value of v w * —*; — « ^ ^ when 

2 2? "t - loc /r 

20. Required the value of ^— , when x = 1. 

1 —(2* — *) 2 2 

.fln*. — 1. 

21. Required the value of , when x = 1. 

1 — a: + log. x 

.flws. — 2. 

22. Required the value of — - — , when x = 0. 

sin. xr i 

Ans. I '. 

% losr. x — ~ (x 1 j 

23. Required the value of ^ —J '-, when x = 1. 

(a? — I) log. # 

.flns. !• 

3 

24. Required the value of- i- , when a? = a. 

(a- — «)2 

3. 

Ans, 2a 2 . 
x \ 

25. Required the value of - , when x = 1. 

# — 1 log. x 

Am. i. 

26. Required the value of— — - — — - — :, when x = a. 

^ a* — 2 a?x + 2ax 3 — a? 4 

Ans. <Tj . 

1 x 

27. Required the value of - — ■ , when x — 1. 

log. x log. # 

^?15. 1. 

_ . , , , ,. x 2 — a 2 ne x . 

28. Required the value of — . tan. — , when x = a. 

1 ar 2a 

4 






a n #) 

29. Required the value of — , when x = 1. 

^ COt. -i- * .T 



2a 



sin. x 



SO, Required the value of : , when x — 0. 

x — sin. x 



Jlns. 1 



THE DIFFERENTIAL CALCULUS, 



CSAPTES VI. 

ON THE MAXIMA AND MINIMA VALUES OF FUNC- 
TIONS OF A SINGLE VARIABLE. 

*(47.) In any function y = Fx let the independent variable take a 
particular value x = a, as also a preceding and succeeding value 
x — a — h and x = a + h, then the corresponding values of the 
function, arranged according to those of the variable, will be 

F(a — /i),Fa, F {d+h) ; 
and if a be such that for any finite value of h, however small, and for 
all intermediate values between this and 0, the middle value Fa 
exceeds that on each side, the value x = a is said to render the func- 
tion a maximum ; but if the middle value continue less than that on 
each side between the same limits of h, the value x = a is said to 
render the function a minimum ;| so that we are not always to un- 
derstand by the expression, maximum value of a function, the greatest 
value such function can possibly take the term being of more com- 
prehensive meaning, applying to every state of the function which 
exceeds its immediately preceding and succeeding state, In like 
manner, the minimum value of a function does not always imply the 
least possible value of such function, but equally characterizes every 
state of the function which is less than its immediately preceding 
and succeeding state. 

Before proceeding to the general method of determining the values 
of .c, necessary to render any proposed function a maximum or a 
minimum, we must premise this lemma : 

If the function F (a + h) be developed according to the ascending 
powers ofh, a value so small may be given to h that any proposed term 
in the series shall exceed the sum of all that follow. 

* The maximum value of any function is that, which is greater than those which 
immediately precede and follow it ; and the minimum value is that, which is less 
than those which immediately precede and follow it. It is proper to observe that 
the same function may have several maxima and minima values. Ed. 

| This definition of a maximum and a minimum is but a slight alteration of 
that given by Dr. Lardner in his Differential Calculus, p. 103, 



64 THE DIFFERENTIAL CALCULUS. 

(48.) Let Ah be any proposed term in the development, and let 

Bh' , Ch / , &c. be those which follow, each exponent being greater 
than that which precedes it. We are to prove that h may be taken 
go small that 

Ah a > h? (B + Chi"? + Bh S -P +&c.) 
Putting S for the sum of the series within the parentheses, it is 

obvious that h may be taken so small that S/i may be less than 

any proposed quantity A, and that therefore if hi be such a value we 
must have 

Ah A > S/i' 3 

which establishes the proposition. As Sti is less than A for 

h = ti, the expression continues less than A for every value of h 
less than ti. 

(49.) Let us now inquire by what means we may determine those 
values of a: which render any proposed function Fx a maximum or a 
minimum. In order to do this, let .r be changed into x ± h, then 
by Taylor's theorem 

y J dx dx" 1 • 2 dx 3 1 • 2 • 3 

dHj h}_ 

Now if x = a render the proposed function a maximum, then there 
exists for k some finite value ti, such that for all the intermediate 
values between this and we have 

Fa > F (a db h), 
and, consequently, 

But if this value render the function a minimum, then, for all the in- 
termediate values of k between h = ti and k = 0, we have 

Fa < F (a ± h) 
and, consequently, 



THE DIFFERENTIAL CALCULUS. 65 

It has, however, been proved above, that a value may be given to 
h small enough to render the first term in each of the series (1) and 
(2) greater than the sum of all the other terms, and that this first term 
will continue greater for all other values of h between this small 
value and 0, so that, for each of these values of /i, the sign belonging 
to the sum of the whole series is the same as that of the first term ; 
it is impossible, therefore, that either of the conditions (1) or (2) can 

exist for both + \-M h and — T-/l /i, unless f-^1 = ; we con- 
L dx dx ^dx 

elude, therefore, that those values of x only can render the function 

a maximum or minimum which fulfil the condition 

ax 
expunging, therefore, the first term from each of the series, (1), (2), 
we have, hi the case of a maximum, the condition 

&^± &>£$ + **'*'• '-<& 

and in the case of a minimum, 

r d 2 v _ h 2 , ,-dhi h 3 , . 

^T^2 + ^TT2T-3 + &C -7 0...(4). 

Now the former of these conditions cannot exist for any of the values 

of h between h = h' and h = 0, by virtue of the foregoing principle, 

d 2 i{ 
unless \-fY~\ ls negative, nor can the latter condition exist unless 

d 2 y 
[ Vy] is positive, that is, supposing that these coefficients do not 
ax 

vanish from the series (3) and (4). 

We may infer, therefore, that of the values of x which satisfy the 

condition — = 0, those among them that also satisfy the condition 
-~ /_ belong to maximum values of the function, while those ful- 

CiX 

d 2 y 
filling the condition -~- ~7 belong to minimum values of the func- 

a dx 2 & 

tion. It is possible, however, that some of the values derived from 
the equation ~ = may, when substituted for x in -r4-, cause this 

* See Note (C"). 
9 



66 THE DIFFERENTIAL CALCULUS. 

coefficient to vanish, in which case the conditions (1), (2), become 
and 

dhi 
which are both impossible unless \_-r~r1 — °> f° r reasons similar to 

OX 

d 11 
those assigned above, and, unless, also [—4-1 Z in the case of a 

maximum, and f-H^rl 7 in the case of a minimum ; that is, on the 
u aar 

supposition that this coefficient does not vanish from the series (5) 

and (6). If, however, this coefficient does vanish, then, for reasons 

similar to those assigned in the preceding cases, the following coeffi- 

d 5 y 

cient ■—— must also vanish, and the condition of maximum will then 

. d?y tfy 

De L~TT] Z 0, and the condition of minimum [yr] 7 0, andsoom 

It hence appears, that to determine what values of x correspond 

to the maxima and minima values of the function y = ~Fx, we 

must proceed as follows i 

dii 
Determine the real roots of the equation -j- = 0, and substitute 

U/X 

dhi dhi 
them one by one in the following coefficients — ^~, — ^-, &c. stopping 

U/X OjX 

at the first, which does not vanish. If this is of an odd order, the 

root that we have employed is not one of those values of x that 

renders the function either a maximum or a minimum ; but if it is of 

an even order, then, according as it is negative or positive, will the 

root employed correspond to a maximum or to a minimum value of 

the function. 

(50.) It must however be remarked, that, should any of the roots 

dy 
of the equation -7- = cause the first of the following coefficients, 

which does not vanish, to become infinite, we cannot apply to such 
roots the foregoing tests for distinguishing the maxima from the 



THE DIFFERENTIAL CALCULUS. 67 

minima, because the true development of the function for any such 
value ofx begins to differ in form from Taylor's development, at that 
term which is thus rendered infinite (4), so that we cannot infer, 
from Taylor's series, whether the power of h, which ought to enter 
this is odd or even. 

In a case of this kind, therefore, we must find, by actual involu- 
tion, extraction, &c. the true term that ought to supply the place of 
that rendered infinite in Taylor's series for x = a. If this term take 
an odd power of A, or, rather, if its sign change with the sign of h, 
then x — a does not render the function either a maximum or a mini- 
mum ; but if the sign does not change with that of A, then the value 
of x renders the function a maximum or a minimum, according as 
the sign of this term is negative or positive. 

To illustrate this case, suppose the function were 

y = b -f- (x — a) 3 

dhj _ 10 _j_ 

dy 
Now the equation -p = gives x = a, so that if any value of x 

could render the proposed function a maximum or a minimum, 
this most likely would be it. By substituting this value of x in 

#y 

j-o- the result is infinite, and we cannot infer the state of the function 
ax 

from this coefficient ; therefore, substituting a ± h for x in the pro- 
posed, we have 

F (a ± h) = b ± ifi 

and, as h 3 obviously changes its sign when h does, we conclude that 
the function proposed admits of neither a maximum nor a minimum 
value. 

Again, let 

y = b -f- (x — a) 3 

dy _ 4 i 

.:-~-- {x -a)> 



68 THE DIFFERENTIAL CALCULUS. 

dry 4 _2 

dJ ~ 9 ( * ~ ft) 3 

The equation ~ = gives x = a, a value which causes -7-^- to 

become infinite ; therefore, substituting a ± h for a: in the proposed, 
we have 

. F (a ± h) == 6 = /^ 

and, as the sign of A 3 is positive whatever be the sign of h, we con- 
clude that the value x = a renders the function a minimum. 

(51.) There remains to be considered one more case to which the 
general rule is not applicable, and which, like the preceding, arises 
from the failure of Taylor's theorem. We have hitherto examined 
only those values of x for which Taylor's development is possible, as 
far at least as the first power of h, but we cannot say that among 
those values of x, which would render the coefficient of this first power 
infinite, there may not be some which cause the function to fulfil the 
conditions of maxima or minima; therefore, before we can conclude 

dy 
in any case that the values of x, deduced from the condition y = 0, 

comprise among them all those which can render the function a 
maximum or minimum, we must examine those values of ^arising from 

dy 

the condition y- = co by substituting each of these ± h for x in the 

proposed equation, and observing which of the results agree with the 
conditions of maxima and minima in (47). 

(52.) If the function that y is of a: be implicitly given, that is, if 

u = F (*, y) = ; 

then, by (39), we have, for the differential coefficient, 

dy du . du 

dx ~ ~ dx ' dy ' ' ' ^ '' 

dy du 

and therefore, when -r- = 0, we must have -5- = ; hence, the 

values corresponding to maxima and minima, are determinable from 
the two equations* 

* Other values may be implied in the condition - - = oo which leads to — - 
J r dx dy 

= 0, but to ascertain which of these are applicable would require us to solve the 

equation for y. 



THE DIFFERENTIAL CALCULUS. 69 



:}. 



u = 

dx 



Having found from these values of x that may render w'a maximum 

or a minimum,* as also the corresponding values of y itself, we must 

dry 
substitute them for x and y in t-j, when those values of y will be 

maxima that render this coefficient negative, and those will be mini- 
ma that render it positive. But those values that, cause it to vanish, 
belong neither to maxima nor to minima, unless the same values 

cause also — ^— to vanish, and so on. 
dx 6 

The second differential coefficient may be readily derived from 

(1 ), for, putting for brevity 

<% = __M 

dx N' 

we have 

dM , dM dy s ^ T t dN , dN dy 

N ( + • — ) — M ( h . — ) 

d 2 y _ K dx dy dx dx dy dx 

_ s __ . 

M 
which, because — = 0, becomes for the particular values of x re- 
sulting from this condition, 

- r d l i _ r d * U "I ^ r^-i ,oN 

[ rf^ ] ~-^ • [ dy^ ' ' ' (3) * 
By differentiating the above expression for -~ we shall find 

r d 3 y .. r rf\t _ . r du n 

*& =-[*?"]-%] • • • (4). 

and so on. 

(53.) Before we proceed to apply the foregoing theory to exam- 
ples, we shall state a few particulars that may, in many instances, be 
serviceable in abridging the process of finding maxima and minima. 

* Gamier, at p. 271 of his Calcul Differential, says, that, by means of the 
equations (2) "on obtientles valeurs de x ety par lesquelles F (x, y) devient ou 
peut devenir maximum ou minimum ;" but this is evidently a mistake, since, by 
hypothesis, F (,r, y) is always == 0. 



10 THE DIFFERENTIAL CALCULUS. 

1. if the proposed function appears with a constant factor, such 
factor may be omitted. Thus, calling the function Ay, the first dif- 
ferential coefficient will be A -X and A -=? = leads to -^=0,also 

dx dx dx 

— — = leads to -— = 0, so that A may be expunged from the 

dx dx 

function. 

2. Whatever value of x renders a function a maximum or mini- 
mum, the same value must obviously render its square, cube, and 
every other power, a maximum or minimum ; so that when a proposed 
function is under a radical, this may be removed. The rational 
function may, however, become a maximum or a minimum for more 
values of x than the original root ; indeed, all values of x which 
render the rational function negative will render every even root of it 
imaginary ; such values, therefore, do not belong to that root ; more- 
over, if the rational function be = 0, when a maximum, the corres- 
ponding value of the variable will be inadmissible in any even root, 
because the contiguous values of the function must be negative. 

3. The value x = go can never belong to a maximum or minimum, 
inasmuch as it does not admit of both a preceding and succeeding 
value. 

EXAMPLES. 

(54.) 1. To determine for what values of # the function 
y — a 4 + b 3 x — c 2 x 2 
becomes a maximum or minimum, 

dx dx** 

From the second equation it appears that, whatever be the values of 

dv 
x, given by the condition -=2- = 0, they must all belong to maxima. 

6 3 , 
From b 3 — 2c 2 x = 0, we get x = — — ; hence 

b 3 , 6 6 

when x = — - .•.!/ — « + tit? a maximum. 
2c 2 4c 2 

dy 

The equation -7- = go would give, in the present case, x = go, a 

value which is inadmissible (53), 



tion 



THE DIFFERENTIAL CALCULUS. 71 

2. To determine the maxima and minima values of the func- 

y = 3aV — b*x + c 5 



putting 



^ = 9aV-6 4 ,^ = 18a 2 ^ 
dx dxr 



b 2 

9aV — 6 4 = 0.-. a;= ± — 

3a 



Substituting each of these values in -~~ we infer from the results 
that 



when x = — . . 
3a 


• 'V 


= c 5 — 


2b 6 

— — , a mm. 

9a' 




6 2 

a; = — — - 
3a 




2/ = c * 


, 26 6 

+ , a max. 

9a 


3. To determine the maxima and minima values 


of the function 




y = 


= V2ax 






Omitting the radical 




du 

' dx 






u = 


2ax .' 


2a, 





as this can never become or oo , we infer that the function has no 
maximum or minimum value., 

4. To determine the maximum and minimum values of the 
function 



y = \Z4a 2 ^ — 2a.r 3 . 

Omitting the radical and the constant factor 2a (53), 

u = 2ax 2 — x 3 , 

du „ _ d 2 u 

..._ = 4a*-3^ =4 a_6* 5 

4a 
.*. x (4a — 3a?) = .•. x = 0, or x = -~-. 

d?u 
Substituting each of these values in - 1 - 5 -, the results are 4a and 

dx 

-~ 4a ; hence 

when x = , . . y = 0, a minimum. 



iZ THE DIFFERENTIAL 


CALCULUS, 


X 


_ 4a 


8 


a 2 , maximum 


If, instead of the preceding, 


the examp' 


e had been 




y = 


- y/2ax~ i — 


4crx 2 , 


we should have had 








du 
dx 


3x 2 — 


dru 


= 6x — 4a. 


.-. x (3.r 


- 4a) 


= 0, .-. x 


— 0, or x = 



4a 
~3~ 

the same values as before ; but the first corresponds here to a maxi- 
mum, since it makes - — negative ; this value, therefore, must, by 

(53), be rejected. If, indeed, we substitute db h for x, in the pro- 
posed function, it becomes 



y = \/ — 4a 2 /r ^ 2a/? 3 , 
where h may be taken so small as to cause the expression under the 
radical to be negative for all values of h between this and 0. 

5. To determine the maxima and minima values of the function 



y = a -\- %/a- — 2a 2 x + ax~. 
If y is a maximum or minimum, y — a will be so ; therefore, trans- 
posing the a, and omitting the radical (53), 
u = a? — 2a 2 x + ax 2 

du n _ , d 2 u 

-j- = — 2a 2 + 2ax, — = 2a, 

dx dx 2 

.•. — 2a 2 + 2a.r = .*. x = a, 

.-. when x = a . . . y = a, a minimum. 

6. To determine the maximaand minima values of the function 



y = 



(a — x) 2 

In solving this example we shall employ a principle that is often found 
useful, when the proposed function is a fraction with a denominator 
more complex than the numerator. Instead of the function itself we 
shall take its reciprocal, which will give us a more simple form, and 
it is plain that the maxima and minima values of the reciprocal of a 



THE DIFFERENTIAL CALCULUS. 



function correspond respectively to the minima and maxima of the 
function itself. Omitting, then, the constant cr, and, taking the re- 
ciprocal, we have 



a 2 — 2ax + x 2 a 2 


— 2a 


X X 


du a 2 d 2 u 
dx x 2 ' dx 2 


_ 2d 2 



— + 1 = .-. x = ± a .-. [ — ] = 



a 

hence x = a makes u a minimum, and x ■ = - — o makes it a maxi- 
mum, therefore 

when x = a . . . y == oo , a maximum, 

a; — — a . . • y — — |- a, a minimum. 

7. To determine the maxima and minima values of the function 

V = & + V { x — a Y° 
Omitting b and the radical 

u = {x — a) 5 
..._=6(«-o/, sr = 4.6(»-.)' 

...5(.r-«.) J = 0.-..x- = «.-. [g-] = 0. 

As this coefficient vanishes, we must proceed to the following, 

which however all contain x — a, and therefore vanish, till we come 

du 
to -=-r- = 2 * 3 * 4 • 5 ; as therefore the first coefficient which does 
dx 3 

not vanish is of an odd order, the function does not admit of a maxi- 
mum or a minimum. 

8 To determine the maxima and minima values of the function 



dii dhi 1 

f x = ,;* (l + log. x), ^- = *■ \~ + (l + log. Xf\. 

10 






74 THE DIFFERENTIAL CALCULUS. 

The factor x r can never become 0, therefore 

(1 + log. x) — .-. log. x = — 1. 

1 

.*. x = e l = — 

c 

\_ 

l 
1 1 7 

.*. when x = — , af = ( — ) , a minimum. 

9. To determine the maxima and minima values of y in the 
function 

u = x* — 3axy -f if = 

^ = 3** — Say.-. (52) 

ar 5 — Saxy -f- ?y 3 = , a^ 2 

3^ __ 3ai/ = 0* •'* 2/ = ~ •*• * G — 2a3 * 3 = ° 

.«. x — or x == a y 2 .-. (52) 

d 3 ?/ rf 2 w dw z 4 2a 2 n 

t?] = -13?] + t^] =-[«*] * [^-3«x] =-[^-.] 

2 2 

or — — = 



a 3/2 — 1 

.-. when a: = . . . . y = 0, a minimum. 

x = a%/2 . . . . y = a %/4, a maximum. 

10. To divide a given number a, into two parts, such that the 
product of the with power of the one and the nth. power of the other 
shall be the greatest possible. 

Let x be one part, then a — # is the other, and 

y = x m (a — x) n = maximum, 
.-. -^ = roaf -1 (a — x) n — nx m (a — x) n ~ l 

(XX 

= x m ~ l [a — a:)"- 1 \ma — (m + n) x\ = 0, 
. *. x = 0, or a — x = 0, or 



THE DIFFERENTIAL CALCULUS- 75 

i 

ma — (m + n) x = 0, 
which give 

ma 

x = 0, x = a, x — ■ . 

m + ft 

The first and second of these values are inadmissible, because the 
number is not divided when x = or when x = a. 
Substituting the third value in 

d 2 y 

_JL r= x m ~ 2 (a — .r) n ~~ J (wia — (ro -f - ft) xy — m (a — x) 2 — nx 2 ^ 
(Xxr . 

we have 

[ £■] = - W— [« - *]r* }- [« - «]■ + ■*! 

which i3 negative because each factor is positive, hence the two re- 
quired parts are 

ma f - na . . 

and — — — being to each other as m to n. 



m + ft m + » 
Cor. If m = n the parts must be equal. 
An easier solution to this problem may be obtained as follows : 

m> 

Put — ■ = p and determine x so that we may have 

u = x p (a — x) = A maximum, 

du 
,»,t- = px*- 1 {a — x) — x p 

= a*" 1 [pa— (p + 1) *J = 0, 

.*. ar = = or »a — (p -\- 1) x = .-. x = — ^ — . 

p + 1 

This last value substituted in 

% = * p - 2 \P* ~ (P + 1) *\ - iP + l ) **-* 

causes the first term to vanish ; the result is therefore negative, so 

- pa ma . 

that x = — V — = ; corresponds to a maximum value of//, 

p + 1 hi -r » 

and therefore (53) to a maximum value of u n = x m (a — x) n . 

Another easy mode of solution is had by using logarithms, for it is 



^ 6 THE DIFFERENTIAL CALCULUS. 

obvious that since the logarithm of any number increase?- with the 
number, when this number is the greatest possible, its logarithm will 
be so also. 

.-. m log. x + n log. (a — x) ~ max. 

du m 11 

'"• til = ° •'• ma — 0» -f n) x = 

ax x a — x K * 



in -f n 
as before. 

The expression for the second differential coefficient is — (m + 
n) showing that the foregoing value of x renders the logarithmic ex- 
pression a maximum. 

11. To divide a number a, into so many equal parts, that their 
continued product may be the greatest possible. 

It is obvious from the corollary to the last example, that the parts 
must be equal, for the product of any two unequal parts of a number 
is less than that of equal parts. 

Let x be the number of factors, then, 

{-) —a, maximum, 
x 

•*• l °g- (~T = -*log. (-) = a maximum, 
x x 

a 

.-.log. 1 = 

X 

a 1 1 ^ a 

.-.-== log. -1 1 = e .•. x = -* 
x e 

hence the proposed number must be divided by the number e = 
2-718281828. 

12. To determine those conjugate diameters of an ellipse which 
include the greatest angle. 

Call the principal semi-diameters of the ellipse a, b, the sought 
semi-conjugates x, x' and the sine of the angle they include y. Then 
(Anal. Geom.) 

* There is obviously no necessity to recur to the second differential coefficient 
to ascertain whether this value render the function a maximum or a minimum, 
since it is plain that there is no minimum unless each of the parts may be 0. 



THE DIFFERENTIAL CALCULUS. 77 



& -f x '2 = a? + fr .-. a-' = s/a 2 + b 
draw = ab ,\ « = — — 



«6 



■'•!( = 



max. 



x\f a 2 + b 2 — x 2 

Omitting the constant ab, inverting the function (ex. 6.) and squar- 
ing, we have 

u = a 2 *? + b 2 x 2 — x* = max. 

# . # -JL = 2a 2 x + %b 2 x — 4x 3 = 0, 
ax 

a 2 -j- b 2 x 2 -\- x" 2 

.-. x = 0, cr + b" — 2x~ == .'. or — = . 

2 2 

The first of these values is inadmissible, from the second we find 
that 

x 2 = x' 2 

hence the conjugates are equal. For the second differential coeffi- 
cient we have 

^ = 2a 3 + 26 2 — 12z 2 
dx 2 



This being negative, shows that x — \f corresponds to a 

maximum value of w, or to a minimum value of y, so that the conju- 
gates here determined, include an angle whose sine is the least pos- 
sible ; and this happens when the angle itself is the greatest possible 
(being obtuse), as well as when it is the least possible. 

13. To divide an angle & into two parts, such that the product 
of the ?ith power of the sine of one part of the »ith power of the sine 
of the other part may be the greatest possible. 
Let x be one part, then & — x is the other, and 

sin. n # . sin." 1 (d — x) = maximum, 
. \ n log. sin. x + m log. sin. (& — x) = maximum, 
n cos. x m cos. (d — x) 



sin. (d 



= 0, 



78 THE DIFFERENTIAL CALCULUS. 

.\ n tan. (6 — or) = m tan. x, 

.'. n : m : : tan. x : tan. {& — x), 

.'. n + m : n — m : : tan. x + tan. (d — x) : tan. x — tan. (6 — x), 

: : sin. 6 : sin. (2x — d),* 

.•. sin. (2a: — < 



which determines x. 

14. Given the hypothenuse of a right-angled triangle to deter- 
mine the other sides, when the surface is the greatest possible. 

Call the hypothenuse a, and one of the sides a*, then the other will 

be >/ a 2 — x 2 and the area of the triangle will be 

x — 

r y/ a 2 — x 2 = maximum. 

.-. u = a 2 x 2 — x A = maximum. 

.•.-7- = 2a~x — 4ar = .•. x = or a: = -7-. 
da: V 2 

Substituting the second value in 

d 2 u 

_ r = 2o ._ 12ar 

the result being negative, shows that the above value of x corresponds 

to a maximum. Therefore the required sides are each -— . 

v2. 

15. To determine the maxima and minima values of the function 

y = x 3 — 18a: 2 + 96a: — 20. 
when a: = 4....?/ = 356a maximum. 
x = 8 .... y = 128 a minimum. 

16. To determine a number a-, such that the 1 th root may be 
the greatest possible. 

Ms. x = e = 2-71828 .... 

17. What fraction is that which exceeds it3 with power by the 
greatest possible number ? 

m-l j 

Jlns. s/ — . 
m 

* Dr. Gregory's Trigonometry, p. 47, Equation (S). 



THE DIFFERENTIAL CALCULUS. 79 

18. Given the equation 

y 2 — 2mxy + x 2 = a 2 , 
to determine the maxima and minima values of y. 

ma a 

When x = — — . . . . y = — — _ , a maximum, 

\f 1 — m 2 y/ 1 — m 2 

— ma — a 



y = 



a minimum, 



V 1 — m 2 VI — »* 

19. Given the position of a point between the sides of a given 
angle to draw through it a line so that the triangle formed may be the 
least possible. 

Ans. The line must be bisected by the point. 

20. The equation of a certain curve is ahj = ax 2 — x 3 required 
its greatest and least ordinates. 

When x = fa . . . . y = maximum, 
x = . . . . y = minimum. 

21. To divide a given angle & less than 90° into two parts, x and 

6 — .r, such that tan. n x . tan. m (6 — x) may be the greatest possible. 

n — m 

tan. (2x — 6) = ; tan. 6. 

n -\- m 

22. To determine the greatest parabola that can be formed by 
cutting a given right cone.* 

SCHOLIUM. 

(55.) It will be proper, before terminating the present chapter, 
to apprize the student that in the application of the theory of maxima 
and minima to geometrical inquiries, care must be taken that we do 
not adopt results inconsistent with the geometrical restrictions of the 
problem. We know, indeed, from the first principles of Analytical 
Geometry, that when the geometrical conditions of a problem are 
translated into an algebraical formula, that formula is not necessarily 
restricted to those conditions, but, in addition to all the possible solu- 
tions of the problem, may also furnish others that belong merely to 
the analytical expression, and have no geometrical signification, f If, 

* It will be shown hereafter that a parabola is equal to f of a rectangle of the 
same base and altitude. 

t See the Analytical Geometry. 



80 THE DIFFERENTIAL CALCULUS. 

therefore, among these latter solutions there be any belonging to 
maxima or minima, they are inadmissible in the application of this 
theory to Geometry. The following example is given by Simpson, 
at art 47 of his Treatise on Fluxions, to illustrate this. 

From the extremity C of the minor axis of an ellipse 
to draw the longest line to the curve. Suppose F to be 
the point to which the line must be drawn, and call the 
abscissa CE,rr then the geometrical restrictions of this 
variable are such that its values must always lie between 
^ the limits x = and x = 26, a and 6 denoting the semi- 
axes. 

By the equation of the curve. 

ef 2 = y 2 = ^r ( 2bx - * G ) 

.-. CF 2 = u = x 2 + tt &bx — x 2 ) = maximum. 
6° 

= 2 (07 - ■ — — -pX) =0 

orb 




d 2 —b 2 

and since -^ = 2 (1 — -^) it follows that the foregoing expres- 
dzr o~ 

sion for x renders u a maximum for all values of b less than a, and a 

minimum for all values of b greater than a. Hence if the relation 

between a and 6 be such that -5 jr 2 may exceed 26, the analytical 

CL O 

expression for CF will admit of a maximum value, although such 
value, not coming within the geometrical restrictions of the problem, 

a?b 

is inadmissible. If the relation between a and 6 be such that ~ r- 

a —— o 

= 26, that is, if a 2 = 2b 2 , the solution will be valid, and in the ellipse 
whose axes are thus related CD will be the longest line that can be 
drawn from C, agreeably to the analytical determination, and the 
solution will always be valid if the axes of the ellipse are related so 

2? 

that — is not greater than 26, which leads to the condition 26 £ 

a 2 — 6 2 

not. greater than a 2 . 



THE DIFFERENTIAL CALCULUS. 81 



CHAPTER VII. 

ON THE DIFFERENTIATION AND DEVELOPMENT 

OF FUNCTIONS OF TWO INDEPENDENT 

VARIABLES. 

Differentiation of functions of two independent variables. 

(56.) Let z = F (x, y) be a function of two independent varia- 
bles ; then since in consequence of this independence, however either 
be supposed to vary, the other will remain unchanged : the function 
ought to furninsh two differential coefficients ; the one arising from 
ascribing a variation to x and the other from ascribing a variation to 
y, y entering the first coefficient as if it were a constant, and x enter- 
ing the second as if it were a constant. The differential coeffi- 
cient arising from the variation of # is expressed thus, — ; and that 

dz 
arising from the the variation of y thus, — ; and these are called the 

partial differential coefficients, being analogous to those bearing the 
same name considered in chapter IV. We have seen, in functions 
of a single variable, that if that variable take an increment, and the 
function be developed, what we have called the differential coefficient 
will be the coefficient of the first power of the increment in that de- 
velopment ; so here, as will be shortly shown, the partial differential 
coefficients are no other than the coefficients of the first power of 
the increments in the development of the function from which they 

dz 
are derived. As to the partial differentials they are obviously — dx 

and — dy and hence we call -=- dx + -y- dy the total differential 
of the function, that is, 

dz = — - dx + — - dy, 
dx ay 

and we immediately see that this form becomes the same as that 

11 



82 THE DIFFERENTIAL CALCULUS. 

given in chapter IY. for the differential of F (x, q) as soon as we 

suppose y to be a function of x, for we then have 

dz _ dz dz dy 

*uar dx dy ' dx' 

as indeed we ought. 

In a similar manner, if the function consist of a greater number of 

independent variables as u = F (x, y, z, &c.) we should necessarily 

have as many independent differentials, of which the aggregate 

would be the total differential of the function, that is 

_ du _ . du du 

du = — dx + — du + -r- ds + &c. 
dx dy J dz 

Hence, whether the variables are dependent or independent, we 
infer, generally, that 

The total differential of any function is the sum of the several 
partial differentials arising from differentiating the fund ion relatively 
to each variable in succession, as if all the others were constants. 

We shall add but few examples in functions of independent varia- 
bles, seeing that the process is exactly the same as for functions of 
dependent variables. 

d (x + y) — dx + dy 
d . xy = ydx + xdy 
j x ydx — xdy 



V f 

ay ax 2 chj — ayxdx 



* Vx 2 + y 2 {x? + y 2 )i 

, x ydx — xdit 
dtan.- - = ?—— — ^ 

y r -r x 2 

d y _ ydx — 3y 2 dy — xdy 
2>if—x (3if — x y 

d . a x b y c z = a x b v c z (dx log. a + dy log. b-\- dz log. c) 

, . x ydx — xdy 2 (ydx — xdy) 
d log. tan. - = — — = — — — 

y 2 • x x 2 • 2x 

y A sin. - cos. - y 2, sin. — 

y y y 

dy* = y x log. ydx + y x ~ l xdy. 
(57.) If the function that x, yiso£z is given implicitly, that is by 
the equation 



THE DIFFERENTIAL CALCULUS. 83 

u = F (x, y, z) = 0, 



then 



but (39), 



du. , , .du. 

* = and \-r\= 0, 

* d y% 



d,v 



du _ du du dz _ 

dw _ du du dz _ ' 

<% <% ^ * di} 

du . du dz s .dm, du dz 

.'. du = (— + -r. -r-) dx + (— + -j- . -j-) dy = 

y dx dz dx' dy dz dy' 

= &*+£** 

Thus : let A* 2 + B?/ 2 + Cz 2 — 1 = 0, 

.-. — = 2Aa?+ 2C« — = 
*dx dx 



du. ,_ . nr ~ dz 

-S=2By + 2Cz- 



.-. du = (As + C^y) cfo + (By + C*^) dy = 0. 

(58.) If u = Fr, « being a function of # and ?/, the two differen- 
tial coefficients are (33) 

du _ du dz du _ du dz 

dx dz ' dx' dy dz ' dy 
and the total differential is, therefore, 

* The brackets are employed here for the same purpose as at (37), viz. to im- 
ply the total differential coefficient derived from 11, considered as a function of a single 
variable. This form it will be necessary to adept whenever u contains, besides a:, 
other variables that are functions of x, provided we wish to express the total coeffi- 
cient with respect to x. No ambiguity can arise from our calling these same coef- 
ficients partial in one sense, and total in another. They are partial coefficients in 
relation to the lohole variation of u, but they are total coefficients as far as that 
variable is concerned whose differential forms the denominator; and it may be re- 
marked here, once for all, that when we enclose a differential coefficient in brack- 
ets, we mean the total differential coefficient to be understood, arising from consi- 
dering the function, whose differential is the numerator, as simply a function of the 
variables whose differentials form the denominator. 

d 



84 THE DIFFERENTIAL CALCULUS. 

T du dz . du dz 

du = — - . -j- dx + — . -=- dy. 
dz dx dz dy 

Now it is worthy of notice, that the ratio of the hvo partial differ- 
ential coefficients is independent of F, so that this may be any func- 
tion whatever. Thus 

du , du du dz . du dz dz , dz 

dx ' dij dz * dx ' dz ' chj dx ' dy 
which is an important property, since it enables us to eliminate any 
arbitrary function F of a determinate function /(.r, y) of two variables. 
We shall often have occasion to employ it in discussing the theory of 
curve surfaces. By means of this property too we may readily as- 
certain whether an expression containing two variables is a function 
of any proposed combination of those variables. For, calling .this 
combination z and the function u, we shall merely have to ascertain 
whether or not the above condition exists, or, which is the same thing, 
whether or not the condition 

du dz du dz _ 
dx dy dy ' dx 
exists. For instance, suppose we wished to know whether u = x* 
+ 2x 2 if + if is a function of z = x 2 + ]} 2 * 
Here 

J = 4,3 + 4xy%^ = My + 4tf; | = 2y, J = 2x; 

^du^dz_du^dz = _ ^ = Q 

dx dy dy dx K J J J \ J J ) 

consequently, since the proposed condition exists, we infer that u is 
a function of x. 

We shall now proceed to apply Taylor's theorem to functions of 
two independent variables. 

Development of Functions of two Independent Variables, 

(59.) In the function z = F (ar, y) suppose x takes the increment 
h, the function will become F (x + h, y), y remaining unchanged, 
since it is independent of x, then, by Taylor's theorem, 

•cw , 7 N . dz , d 2 z h 2 d 3 z h 3 

F(* + ft, j) = , + _&+_.__ + _. __ + 

&c. . . . (1). 



THE DIFFERENTIAL CALCULUS. 85 

But if 1/ also take an increment fc, then z will become 

, dz _ . d 2 z ¥ , d?z ¥ 

so that in the expression (1) we must for — substitute 

dz d 2 z d?z 

dz ^ ' djj ~df k 2 d ' dif ¥ 

dx + dx k + dx ' 1 • 2 + dx ' 1 • 2 • 3 + &C * 

dz 2 + d^ ■ + ^ 1 • 2 "*" d* 2 1 -2 • 3 ~ t ~ KC ' 



f ° r ^ 



cfe d 3 z d 



,73 J3 ^3 

d 3 z a ' du d • efy 2 fc 2 rt * dif ¥ 

dx 3 ^ dx* K ^ dx 3 • 1 • 2 T dx' * 1 -2 • 3 ^ 0£C ' 

and so on. Before, however, we actually make these substitutions, 
we shall, for abridgment, write 

dz d 2 z 

d 2 z d ' dj _d?z__ d ' df d***z 

dydx 01 dx ' dif dx l dx => ? dy q dx p 

d?z 

d p • J7 
- d, f 

for 



dx* 

this last expression implying that after having determined the ^th 
differential coefficient of the function z relatively to the variable y, 
the pth differential coefficient of this is taken relatively to the other 
variable x. Hence, the result of the proposed substitutions in (1) 
will be 

F (* + h, y + *) = 



86 



THE DIFFERENTIAL CALCULUS. 



dx 



dz 
dy 



h 



+ 



d 2 z 


h 2 


dx 2 


1 • 2 


d 2 z 






hh 


dydx 




d 2 z 


h 2 



df 



1 • 2 



+ dx 3 

d 3 z 

dy dx 2 
d*z 

dy 2 dx 
i 3 z 
!5 



v 



The general term of the development being 
d q + p z h q h p 



1 -2-3 

hh 2 

1 • 2 

Vh 

1 -2 
1-2-3 



+ &c. 



<%* eta?. (1 -2 . . . 5)(1 -2. . .p) 
If in the proposed function z = F (#, ?/) we had supposed ?/ to vary 
first, then, instead of (1), whe should have had 

d 3 z k 3 



+ &c. . . . (2). 
But, if x take the increment h, 2 will become 



+ 



,dz d 2 z h 2 

dx dx 2 1-2 



+ 



h s 



dif * 1 • 2 • 3 



+ &c. 



dx* 1-2-3 
and, therefore, we must substitute, agreeably to the foregoing nota- 



tion for 



dz 



for 



dy dxdy 
d 2 z 



h + 



d 3 



d*z 



h 



dx 2 dy 1-2 dx 3 dy 1-2-3 



for 



df 

*» 1 *« ft .|. d ** 

dy 2 dx dy 2 dx 2 dy 2 

d 3 z 



+ -7 



d 3 z 



1 • 2 e^cfy 2 1-2-3 



+ &c. 



+ &c. 



dif 



+ 



d 6 



df dx df . dx 2 dif '1-2 dx 3 dy 3 * 1 • 2 • 3 



+ &c 



and so on ; so that the development would be 
F (* + h,y + k) = 



THE DIFFERENTIAL CALCULUS. 



87 



dx 



dz 

dx 



k 



+ 



d 2 z 


h 2 


dx 2 


1 . 2 


d 2 z 






hk 


dxdy 




d 2 z 


k? 



df 



dx* 

d 3 Z 

dx 2 dy 
d 3 z 

dx dy 2 



dif 



1 "2-1 

h 2 k 

1 • 2 
hk 2 

1 • 2 



1-2-3 



+ &c. 



h p . A? 



the general term being 

d p+9 z 

~dx p dif ' (1-2. . .p) (1 -2 .. . g)* 
As this development must be identical with that exhibited above, 
we have, by equating the like powers of h and k, 



d 2 z 



dijdx dxdij dy dx 2 



_d?^_ 
dx 2 dy 



and generally 



d q+p z 



d p+q z 



dif dxP dx p dif 
we conclude, therefore, thatif we first determine the qth differential 
coefficient relatively to the variable y, and then the pih differential 
coefficient of this relatively to the variable a-, the final result will be 
the same as if we first determine thepth differential coefficient rela- 
tively to x, and then the qth differential coefficient of this relatively 
to y ; so that the result is the same in whichever order the differen- 
tiations are performed. 

(60.) We see from the foregoing development, that the partial 
differential coefficients of the first order are the coefficients of h and 
k, the first power of the increments, so that the term containing these 
first powers is in this respect analogous to that containing the first 
power of the increment in the development of functions of a single 
variable, and, by a very slight transformation, it will be seen that the 
same analogy extends throughout a 1 the terms of the two develop- 
ments. For the development just given may be put under the form 
F (x + h, y + k) = 






,,dz dz 

+ ^ h + d^ k) 



88 THE DIFFERENTIAL CALCULUS. 

2 c£r ota « y dy 2 

+ &c. 

where the partial differential coefficients in each term are identical 
with those which appear in the differential of the preceding term, as 
the actual differentiation shows, thus : 
dz _ , dz , 

the coefficients -=-• — , being functions of a: and «, we have 
cte at/ J 

dz _ cl 2 2 d 2 s 

c/j; a^ J dxdy 

_ cfc d 2 2T t cPz 

cZ . -y- = -7—7- eta + — - dy 
dy dydx dy 

and, consequently, 

In like manner, these coefficients being functions of x and y, we 

have 

d 2 z _ d'z d*z 

dx 2 dx i dx 2 dy 

d 2 z _ d 3 z d*z 

dxdy dx 2 dy dx dy 2 



d 3 z dz d 3 z 

dif dif dx dif J 



so that 



d3 *=S^ 3+3 ^^ + 3 w^ + 
w df ■ ■ ■ ■ (3) ' 

and so on ; the numeral coefficients agreeing with those in the cor- 
responding powers of the expanded binomial. 

(61.) Having now applied Taylor's theorem to functions of two 



THE DIFFERENTIAL CALCULUS. 89 

Variables, we may equally extend Maclaurin's Theorem. For, if in 
the foregoing development, we suppose x and y each = 0, the de- 
velopment will become that of the function F (ft, k) according to the 
powers of h and k ; or, substituting x and y for the symbols h and Jc, 
since these are indeterminate, we have 

■l*^ * + £]/> + *■- 

The principles by which we have thus extended the theorems of 
Taylor and Maclaurin are sufficient to enable us to extend these 
theorems still further, even to the development of functions of any 
number of variables whatever, but this is unnecessary. It maybe 
remarked, however, that if we wish to develop a function of several 
variables according to the powers of one of them, it may be done 
independently of any thing taught in this chapter ; for, if all the varia- 
bles but this one were constants, the development would agree with 
that already established for functions of a single variable, and, as 
these constants may take any value whatever, they may obviously be 
replaced by so many independent variables. We shall give one 
instance of this extension of Maclaurin's theorem to a function of two 
independent variables, choosing a form of extensive application and 
of which the development is known by the name of 

Lagrange's Theorem, 

(62.) The function which we here propose to develop according 
to the power of x, is 

u = Fz, in which z = y + xfz, 
z being obviously a function of the independent variables x and v. 
We shall first develop z = y + xfz according to the powers of x : 
this development is by Maclaurin's theorem 

* = W + ^ + [£]^ + [g0^ + & <, 

and if we denote according to the notation of Lagrange the successive 
differential coefficients of/z, relatively to x hyf'z,f"z,J , "z, &c. we 
shall have 

12 



90 THE DIFFERENTIAL CALCULUS. 

* , = y + , .*fk 
d 3 



• -^■+^— B@,s+e-.S>+>-s 



<3 



cfo 3 

&c. &c. 

Consequently, when x — 0, 

&c. &c. 

Hence 

Z V ^ XjZ y^tt'i^ d y Vl:. : 2 + dtf *l-2.3 

+ &c (1). 

Now, instead of this development, we should obviously have 
obtained that of Fz = F (y + xfz), if in place of z and its differen- 
tial coefficients we had employed Fz and its differential coefficients. 
We should then have had 
u = F {y + xfz) . . therefore . . [ u ] = Fy 

du _ du dz r^!i-i = ^El f 

~dx"~~dz % ~dx *~ dx l— dy JV 

d?u_d 2 u dz 2J , da <Pz_ d 2 u d 2 Fy 

d?~lz j[ dx ) dz' dx 2 ' ' * ' l da» A dy 2 Uy) " r 

dFy 
dFy d.jfyf _ d '~d^^ )2 
dy ' dy dy 

&c. &c. 

Hence 



THE DIFFERENTIAL CALCULUS. 91 

-V !L -^ + *' (2) ' 

and this is Lagrange's Theorem.* 

From this remarkable expression, which includes that marked (1), 
other forms may be readily deduced as particular cases. Of these 
the two following are the most important. 

Put x == 1, then the formula (1) becomes 

JTJ J JJ-r dij 2< 2 -r ^ 1-2-3 

&c (3), 

and the formula (2), 

A* *^W , 

« = F (9 +/z) = Fy + - dy J f,j + ^— . — 2 + 

df • TT1 + &c W- 

(63.) We shall terminate the present section with one or two ex- 
amples of the application of these formulas, referring the student for 
more ample details on this subject to Lagrange's Resolution des 
Equations Numeriques, note xi. ; and Jephson's Fluxional Calculus, 
vol. i. 



EXAMPLES. 

1. Given z 3 — qz + r = 0, to develop z according to the pow- 
ers of r. 

Since here z = - -\ , we have » — -,fz = -2?.: fy = - (- ) 3 

q q q J q JJ q K q } 

* For another and very complete demonstration of this theorem see note (B) 
at the end. 



92 THE DIFFERENTIAL CALCULUS. 

<fy f ' dy q 2V ' dif q 3 ' dy 2 (f ' % 

Hence, by the formula (3), we have, by putting for y its value - 

r , 1 r 3 6 r 5 , 8- 9 r 7 

2 = h — . . ' . — 4- &c. 

<j q q" nr l'2cf g J ^l-2-3^ g 7 



f ._ . r 



2 A„,4 Q . Q.,6 



6r 4 8 • 9r 



2. Given the radius vector of an ellipse, viz. (Jlnal. Geom.) 

1— e 2 



r = a 



l-\- e cos. w 
to develop r n , according to the powers of cos. w. 

Since r = a (1 — e 2 ) — e cos. o . r, we have, by putting */ for 
a (1 — ■ e 2 ) and # for — e cos. w, 

Fr = F (y + z/r) = F (y + x • r) = (y + js • r) n . 
Hence, by the formula (2) 

d.^-y 2 
dy n x dy u x 2 

'" = r + ify-T + —j r -vr2 + 

_d'J__ X s 
ihf ' 1-2-3 + &c - 

x x 2 

= if + ny».- + n(n+l)y n . — - + 

n(n+ l)(n + 2)f. T ^+ &c. 

= a" (1 — e 2 )" (1 - + — e 2 cos. 2 a — 

n(n+ l) (n + 2) 3 / 

— — e 3 cos. 3 w + &e. 

3. It is required to revert the series 

a + Pz + yr + W + &c. = 0, 



THE DIFFERENTIAL CALCULUS. 93 

that is, to express the value of z in terms of the coefficients. Here 

z = -^--^( 7 + Szi-Szc.)=y+fi 
p P 

therefore, by the formula (3), 

* = y — j3(r + ^ + &c -)+ dy '~2~ 

d 2 - J (7 + ^ + &c.) 3 x 

+ &c. 



cty 2 * 1 • 2 • 3 



|1 (y + fy + &C.) (« + 2S7/ + &c.) 
— 5 37 (7 + % + &c ') 3 + &c - 



7 



2/ 2 - 


"1 


3 S 4 


— &c. 




+ 


2y 2 


V 4 


+ &c. 
+ &c. 



+ &c. 
where y == -=- . consequently 

4. Given 1 — z + az = to develop log. z, according to the 
powers of a. 

log. z = a + i a 2 4- i a 3 + £ a 4 + &c* 

* This we know from other principles ; for, since the proposed expression re- 
duces to z = .*. log. % = — log. (1 — a) and this, in the hyperbolic system, 

is equal to the above series. (See the Essay on Logarithms, p. 3.) 



94 THE DIFFERENTIAL CALCULUS. 



CHAPTER VIII. 

ON THE MAXIMA AND MINIMA VALUES OF FUNC- 
TIONS OF TWO VARIABLES, AND ON CHANGING 

THE INDEPENDENT VARIABLE. 

(64.) It remains to complete the theory of maxima and minima 
by applying the principles established in Chapter VI. to functions of 
two independent variables. 

The same character belongs to a maximum or minimum function 
of two variables that belongs to a- maximum or minimum function of 
one variable, that is, the maximum value exceeds the contiguous va- 
lues of the function, and the minimum value falls short of them. 

Hence, if 

z = F (x, y) 
be any function of two variables, which becomes a maximum for cer- 
tain particular values of them, then h and k being finite increments, 
however small the condition is that, between such finite values and 
0, we must always have 

F [>, y-] > F [x ± h, y ± &], 
and, consequently, (60), 

If, therefore, of the small values which we suppose h and k to take, 
h be the smallest, a part of k may be taken so small as to be less than 
h, or, which is the same thing, equal to one of the values of h between 
the proposed value and 0, so that we have h' = k' ; therefore, the 
above condition is 

dz dz , i d 2 z d 2 z d 2 z t2 

*- dx dx 2 dx 2 , dx dy dy 2 

This condition being similar to (1) art. (49), we infer, by the same 

reasoning, that 

dz dz _ 

dx dy 

* This is the manner in which analysts have agreed to express an isolated ne- 
gative quantity ; which must necessarily have resulted from the subtraction of a 
greater from a less quantity. It is not, however, to be inferred that a negative 
quantity is less than zero, as the above expression indicates, as such supposition 
would be manifestly absurd. Ed, 



HxF±2_ +xr ]^+&e.<0, 



THE DIFFERENTIAL CALCULUS. 95 

which cannot be for both the signs ± unless 
dz _ dz _ 
eta ' tty 
By continuing to imitate the reasoning in (49), we find that these 
same conditions must exist for all the values of the variables that ren- 
der the function a minimum. 

Hence (49), we have, in the case of a minimum, the condition 

.d 2 z d 2 z , d 2 Z 

± 2 + — 

da? dxdy dy 2 

and in the case of a minimum, 

so that, supposing these first terms do not vanish for the values of x 
and y given by (1), the condition of maximum is 
d 2 z d 2 z d 2 z 

'-dx 2 dx dy dy 2 

and the condition of minimum, 

d 2 z d 2 z d 2 z 

*-dx 2 dy dx dif 

In either case, therefore, the expression within the brackets must 
have the same sign independently of the sign of the middle term. To 
determine upon what other condition this depends, let us represent 
the expression by 

A ± 2B + CorA(l ± 2~ + -£). 

A. A. 

T»2 T>2 

Adding — — = to the quantity within the parenthesis, its 

A A 

form is 

Now this expression will always have the same sign as A provided 

C B 2 

C has, and that — > — 2 , that is, AC J B 2 or AC — B 2 7 0, be- 
cause then the factor of A will be necessarily positive. Hence, be- 
side (1), the condition that a maximum or a minimum may exist is 



96 THE DIFFERENTIAL CALCULUS. 

r d 2 z d 2 z , d 2 z NO _, 

and we are to distinguish the maximum frcm the minimum by ascer- 
taining whether the proposed values of x and y render 

or, which amounts to the same, whether 

^-ZOor/0, 

d 2 z d 2 z , 

smce -=-— and -r-r- have the same sign. 
dx 2 dif 

Should any of the values determined from (1) cause the coefficient 
of h' 2 to vanish, there will be no maximum or minimum for those va- 
lues unless the coefficient of the following term vanishes also. 

EXAMPLES. 

(65.) 1. To determine the shortest distance between two straight 
lines situated in space. 

Let the equations of the two lines be 

x = az + a ) j f x' = a'z' + a' /1N 

,=A,+ / 8-J« d U' = lV+- j S (1) 

then the expression for the distance between any two points {x, y, z) f 
(.r' y' z') is (Ancd Geom.) 
1)2 = u = ( X — x'f + (1/ — xj'f + (z — z') 2 ■ 

= (a—a+az—a'z'y+((3 — (3' + bz—b'zy + (z—zy 
and this expression, containing the two independent variables z, z' is 
to be a minimum. Hence by the condition (1) 

— =2(z--z')+2a{a--*' + az--a f z')+2b{(3-P' + bz-b'z) = o) 

dz \. (2) 

^=-2(z--z')+2aXa--* f +az--a'z')+2b\(3-(3'+bz-b'z)=0\ 
dz J 

and from these equations the proper values of z, z' may be readily 
determined, which substituted in the expression for u, render it the 
least possible. That these values really belong to a minimum is evi- 
dent, because, 



THE DIFFERENTIAL CALCULUS. 97 

= — 2 (1 -f aa' + 66')' 

. _ . . <Fm d 2 « . . . . , iL , 

and this proves that -— , -pj are both positive, and that 

dz T 'd^~^dxly ) ' 

Since the equations of a straight line passing through two points 
(x, y, z) t (a/, y', z') are 

x — x' = a'' (z — 2') ) 

y-y' = b"( z -z')i 

we have, by substitution, when these points are on the lines (1) 

a — a! + az — a' z' = a" (z — z r ) ) ,~x 

(3 — (3'+bz — b'z' = b"(z — z')]' ' ' ' ^ 

hence, if this line be that in question, we have, by combining the 
equations (2) with these, the conditions 

1 + aa" + bb" = 0, 1 + a a" + b' b" = 0, 
which conditions shew, that this minimum straight line is perpendicu- 
lar to both the lines (1). (See Anal. Geom.) From these conditions 
we get 

a" = h '~ h b" = - a '~~ a 

a'b — ab n a'b — ab' ' 

by means of which, and the equations (3), the expiession for D be- 



(*-*') 



D = Cl ' b _ ab' V ^~ a# >' + < 6 - b ' r + ( a ' b - ah '? 
in which, if we substitute the value of z — z deduced from (2), we 
obtain, finally, 

(b — b') (a — a') — (a — a ') ((3 — (3') 
V{a — a'Y+ (b — b') 2 +(a'b — ab'f" 
If the numerator of this expression vanish, w T e shall have D = ; 
so that, in this case, the lines will intersect. Indeed, the condition 
of intersection of the two lines, (1), we know (Anal. Geom,) to be 
{b -b') (a-*') = (a- a) (/3 - (3'). 
13 



THE DIFFERENTIAL CALCULUS. 



2. Among all rectangular prisms to determine that which, having 
a given volume, shall have the least possible surface. 

Representing the three contiguous edges of the prism by x, y, z> 
and the volume by a 3 we have 







u 


= 2xy + 2xz + 2yz 


= minimum. 


but since 






xyz = a 3 .*. z — 


_ « 3 
xy 






.'. u 


, 2 « 3 , 2a 3 
= 2xy -f- ■+■ 

y * 


= minimum. 


therefore 


we 


must have the conditions 










du du 

dx 9 dy 






that is, 

2a 3 2 a 3 

2y = - ir = 0,2x = 

a x 2 y 2 

from which we obtain 

y = x = a .•. z = a. 

If these values really correspond to a minimum they must fulfil 

the conditions 

r d?u n — n r d 2 u n -, r d 2 u d 2 u , d 2 u v „_ -, _ 

fe3 7o )[ ^ r ]7.o ) [^.^-(^)T7o, 

and these conditions are fulfilled, since 

l dx 2 J ' l dif J ' l dxdy j 

Hence the required prism must be a cube. 

In the preceding example we might have concluded, without re- 
curring to these conditions, that the results obtained belong to the re- 
quired minimum, there being obviously no other maximum or mini- 
mum, except that which belongs to x = 0, y = 0, z — oo , these 
being the values which cause the differential coefficients to become 
infinite, (see art. 50.) 

3. To divide a given number, a, into three parts, such that the 
continued product of the with power of the first part, the nth power of 
the second part, and the fth power of the third, may be the greatest 
possible. 



THE DIFFERENTIAL CALCULUS. 99 

m, , ma Ua P a 

1 he three parts are • ■ — , ; ; — -, ; ; — so 

r m + n + p m + n + pm-rn-tp 

that the three paits are to each other as the exponents of the proposed 

powers. 

4. To determine the greatest triangle that can be enclosed by a 

given perimeter. 

The triangle must be equilateral.* 

On changing the independent variable. 
(66.) It is frequently requisite to employ the differential coefficients 

7 70 

~ 9 —^- &c, in which x is considered as the principal variable under 
ax dx A 

a change of hypothesis, x, and consequently y being assumed as a 

function of some new variable t. 

It is therefore of consequence to ascertain what changes take place 

in the expressions for these coefficients in such cases. This we may 

do as follows : 

Since according to the new hypothesis 

. y = ¥x and x = ft 

therefore (33) 

dy dy dx dy dy , dx (dy) 

dt ~dx~'~dt ''dx~"dt * dt ~~ (dx) ' 

dv dx 

where for brevity (dy) is put for -— and (dx) for -=-. 

<Py_ _ dry_ d 2 y dy ^ d z x 

dt 2 dx' 2 ' di 2 dx ' dt 2 

. d2 y _ ( d ii (%) ,- , _. rj ,x. _ (<*W*)-(**)(<fr) 
"1? { dt 2 (dx) * {(ix) • { ] (dxy ' 

In a similar manner we might, if necessary, find the expression for 

d 3 y 

-—. It appears, therefore, that 

dy _ (dy) 



dx (dx) 

* For more examples the student may refer to Jephsorts Fluxional Calculut, to 
Garnier's Calcid Differ entiel, or to PuissanVs Problemes de Geometric. 



100 THE DIFFERENTIAL CALCULUS. 

dhj_ = (dhj) (dx) - ( fPx) (r/y) 
dx 2 (dx)* 

&c. &c. 

Ift = y the hypothesis requires that y be considered as the princi- 
pal and x as the dependent variable. In this case 

(«W =^ = I- (*») = 0, &c. (dx) = g, (A) = -^, &c. 

. rf y _ ! _ ! 

eta dj? {dx) 

di 

d-T 2 (oV) 2 

&c. &c. 

These formulas will be brought into use in the second section. 



CHAPTER IS 



ON THE CASES IN WHICH TAYLOR'S THEOREM 

FAILS. 

(67.) It has been shown, in Chapter II., that the general devel- 
opment of the function F (x + h) always proceeds accordingto the 
ascending positive powers of/?, and the principle upon which this 
fact has been established is this : viz. that F.z and F (x + h) must 
necessarily contain the same number of values ; or in other words, 
the sams radicals that enter Fx must also enter F (x-{-h) but no others . 
Hence we might extend the proposition established in (4), and say, that 
not only the general development proceeds according to the increas- 
ing positive powers of/?, but also every particular development, pro- 
vided the particular value F (a -\- h) contain the same radicals as 
Fa, and no more ; and provided, moreover, that Fa is not infinite, 
which we have seen it must be for the true development of F (a + h) 
to contain a negative power of /i, or a log. h, a cot. h, &c. 

(68.) As F (x + h) must contain the same radicals as Fa: and no 



THE DIFFERENTIAL CALCULUS. 101 

others, it follows that F (x + h) — Fx, must contain the same radi- 
cals as Fx. Now, as multiplying or dividing an expression by any 
rational quantity, can neither introduce nor destroy radicals in that 

• r *u- * a • c F (* + /l ) — F * i u 
expression, we infer that the expression tor , h be- 

lb 

ing rational, must contain the very same radicals as Fx, and no others, 
whatever be the value of h ; but when h = 
F (x + h) — Fx_ dFx 
h dx 

hence the first differential coefficient must contain the same radicals 
as the function Fx, and, by the same reasoning, the second differen- 
tial coefficient must contain the same radicals as the first ; conse- 
quently the same radicals must enter each differential coefficient that 
enter into the original function. If, therefore, there be given to x such 
a particular value a, that anyone of the expressions that may be 
under radicals in Fx may become 0, that radical will of course vanish 
from the function, and consequently from its differential coefficients. 
But if a + h be substituted for x instead of a, the same radical will 
necessarily be preserved in F {a + /i), although it will still vanish 
from Fa and the differential coefficients. It follows, therefore, that 
F (a + h) will have more values than 

_ . r dFi_--. r d 2 Fx^ h 2 , B 
F fl +[_]A + [ _]_+ &c , 

so that this cannot be the true development of F (a -f- h). It is easy 
to explain why, in such cases as this, one of the coefficients and in- 
deed all that follow this, must become infinite, for x = a. For the 
exponent of the radical which vanishes for this value, is diminished 
by unity at each differentiation, and being fractional the expres- 
sion under it will at length appear with a negative exponent, and will 
continue to have a negative exponent in all the succeeding coeffi- 
cients ;* so that these, when a is put for x, become infinite. We 
have observed above, that the failing cases of which we are speaking, 

* This does not invalidnte the previous assertion, that the same radicals enter 
the coefficients, that appear in the original function ; for the radical still remains 
however we increase or diminish the fractional exponent by integers, for 
i 

P = ^ fa 



102 THE DIFFERENTIAL CALCULUS. 

arise from the circumstance of x = a causing an expression to disap- 
pear, which is under a radical in Fa?. The student must not confound 
this disappearance of a radical, with that which may arise from a fac- 
tor by which it is multiplied becoming for x = a, for though a 
radical may disappear in this way from Fx, it will not disappear from 
all the differential coefficients, and therefore Taylor's development 
will hold. Thus if a radical in Fx is multiplied by (x — a) m , m 
being a positive whole number, this radical will disappear when x = a, 
but in the mth differential coefficient the factor will be Qb — a) m ~ m , 
which does not vanish when x = a, but becomes = 1, and thus the 
radical with which it is connected will appear in this coefficient. 

We conclude, therefore, that there are but two classes of values 
for which Taylor's development fails ; 1° those which, put for x, 
render Fx = co ; that is, those which are roots of the equation 

— — = : and 2° those which substituted for x, cause an expression 
Fx 

under a radical, to vanish from Fx, and not from F (x + h). To 
this latter class belongs the value x = a for every function containing 
y/ x — a, for the value x = a causes this radical to vanish from Fx, 
but in F (x + h) it enters as y/h. 

(69.) In order to examine these cases more completely, let in 
general 

F(a + h) = A + Bfc+Cfc a +D7i 3 + Mfc B +Nfc B+ * + &c. .(1) 

represent the true development of F (x + h) for x = a, in which 

n -f- - denotes a fraction falling between the numbers n and n + 1. 

P 

We shall show that the n + 1th differential coefficient derived from 

the function Fx becomes infinite for x = a, as also all that follow 
this, but the preceding coefficients are all finite. 

Since in the development (1), h has no fixed value, we may dif- 
ferentiate relatively to h, and we shall have, 

dF(a+/l) =B + 2C/H-3D/i 2 + +M 1 */i»- 1 +N 1 fc" + i- 1 + &c 

dh 

gF( a +fr) =2C + 2 . 3D/l+ +Mz fc-*+N > fc"+?- l .+ &c 

dh 2 

&c. &c. 

* For brevity M„ will be here used to denote the coefficient of h n ~ l in the nth 
differential coefficient derived from Mh n . 



THE DIFFERENTIAL CALCULUS. 



103 



Now, by (30), these several differential coefficients are the same as 

r dF(x + h) r cPF (x + h) 

1 dx J ' L d? J ' ^ 
the brackets denoting the values when x — a. 
Hence, by substitution, in the foregoing equations, 

+ &c (2) 

d?F(x+h) c 2 . sm M + N fc . + I 

L eta 2 J 

+ &c (3) 

&c. &c. 

Putting, now, h = in the equations (1), (2), (3), &c. we have 
the following results. 

Fa = A 



1 

— -2 



da* 
da; 



[S = ^c 



S—D 



[ 5? ] = MJl "~* + N »° n ~" + * + &c - = M * 

r d n+ 'Fx n _ T * , . N„ +1 

C-3S«-3 =N n+1 0^ ^ + &c. = -^L = oo, 

&c. &c. 

all the succeeding differential coefficients being obviously infinite, 

because the exponent 1, which is already negative, continually 

diminishes by unity. 

(70.) It follows, therefore, that if the true development of F(x -\-h), 
arranged according to the increasing exponents of /i, contain for 
x = an fractional power of h, comprised between the powers h n and 






104 THE DIFFERENTIAL CALCULUS. 

h n+l , then the several terms of this development will be correctly de- 
termined by Taylor's theorem, as far as the term containing h n in- 
clusively ; but the terms beyond this become infinite, and therefore 
do not belong to the true development. 

If a term, in the true development of F(a + /*), contain a negative 
power of h, this should be the leading term, as the arrangement is 
according to the increasing exponents ; therefore, this first term, 
when x = a and h = 0, must be infinite, and consequently all the 
differential coefficients, (2), (3), &c. must be infinite. 

(71.) The converse of these inferences are true, viz. 1°. If, in 

d n + x Fx 
the general development of F (a? + h), the coefficient /+i is the 

first which becomes infinite for a particular value of x, then, in the 
true development, arranged according to the increasing exponents of 
/?, the term immediately succeeding that which contains h n , will con- 
tain a fractional power of h, the exponent being between n and n -\- 1. 
For it is obvious, that in order that the n + 1th may be the first of 
the coefficients (2), (3), &c. which contain a negative power of /*, 

h n+ p must be the first fractional power of h which enters the develop- 
ment (1), n + - being between n and n + 1. 2° If, for a particu- 
lar value of x, the function ~Fx become infinite then will all the dif- 
ferential coefficients become also infinite, and the true development 
will contain a negative power of h, or else a log. h, a cot. /*, &c. 
For, if the true development of F(a + h) did not contain a negative 
power of /i, nor a log. h, a cot. h, &c. Fa, which this becomes when 
h = 0, could not be infinite ; hence, such a function of h must enter, 
and therefore, as shown above, all the differential coefficients become 
infinite, for x = a. It is, therefore, necessary to examine the func- 
tion Fa, before we deduce the coefficients from Fx. 

(72.) To obtain the true development of the function for those 
particular values of the variable, which cause Taylor's theorem to 
fail, the usual course is to recur to the ordinary process of common 
algebra, after having substituted a + h for x in Fx. 

Suppose, for example, the function were 



Fx = 2ax — x 2 + a V x 2 — a\ 
and that we required the development of F(x -f- h), for x = a. 



THE DIFFERENTIAL CALCULUS. 105 

Taking the differential coefficient, we have 

dx y/x 2 — a 2 *~dx 

&c. &c. 

As, therefore, the first differential coefficient becomes infinite for 
the proposed value of x, we conclude that the true development of 
the function for that value, when arranged according to the increasing 
-exponents of /?, has a fractional power of h in the second term, the 
exponent of this power being between and 1. 

Substituting, then, a -f h for x in Fa 1 , we have 



F(a + h) = a 2 — h 2 + a \/2ak + h? 



= a 2 — h 2 + aW{2a + hy 



Developing (2a + h) 2 by the binomial theorem, we have 

11// h 2 

(2a+h) 2 =(2a) 2 + — 7 j+ &c 

2 (2a) 2 8 (2a) 2 
consequently, 

3. 5. 

F (a + h) = a 2 + a (2a)*** + — h 2 — - + &c. 

2 {2a)i 8 (2a)f 

Again let the function be 

~Fx -= y/ x + (x — a) 2 log. (x — a), 
and let it be required to find the development of F(a + h). Here 

Fa = V a -f X qo. 

It becomes necessary, therefore, first to ascertain whether this ex- 
pression is infinite ; for, if it be, we are not to proceed with the differ- 
entiation, but to infer, agreeably to art. (70), that the proposed func- 
tion, and all the differential coefficients, become infinite for x = a 
and that consequently the true development contains either a nega- 
tive power of /i, or a logarithm of h. Now, by the method explained 
in (44), we find that, when x = a, the true value of 

(x — a) 2 
(x — a) 2 log. (a? — a) = — 



log. (x — a) 
is infinite. Hence the development of F (a + h) contains log. h, for 

14 



10(5 THE DIFFERENTIAL CALCULUS, 

we readily see that no negative power of h can enter. Substituting 
a -f h for x, in Fx, we have 

F(o + h) = (a + W* + tf log. & * 

SCHOLIUM. 

(73.) In the preceding remarks on the development of functions 
for particular values of the variable, we have said nothing about the 
values of/?, the increment of that variable, having indeed considered 
that increment as indeterminate, or rather of arbitrary value. It 
must, however, be observed that, although the particular value which 
we give to x does not, in any case, fix the value of h, it may neverthe- 
less fix the limit between which and all the values given to h must 
be comprised, in order that for particular values of x, Taylor's de- 
velopment may not fail. This fact is very plain, for if the develop- 
ment holds for all values of x from x = a up to x = b, but fails for 

* The above example is from Lagrange, (Calcul des Fonctions, p. 75,) who has 
given a faulty solution of it, which however is copied by Gamier and other writers 
on the Calculus. The solution here objected to is as follows : 

"Soit 

fx = •/ x -f (x — a) 2 log. (x — a) 
en aura ces fonctions derivees 

fx = — — + 2 (x— a) log. (x — a) + x— a 
2yx 

J 8*V» *— a 

&c. \ 

Si on fait x = a, lafonction secondef'x devient infine, ainsi que toutes les sui- 
vantes. 

"Ainsi le developpment def(x -j- h) par la formule generale deviendra fautif 
dans le cas de x = a, et il contiendra 7i6cessairement le terme /i 2 log. /Y." 

This solution is faulty, inasmuch as it is assumed that f'x is the first derived 
function that becomes infinite for x = a, whereas fx and /a; are also infinite ; but 
a greater fault is, that this process does not lead to the true conclusion, for the 
inference in italics does not follow from it, but this, viz. that the sought develop, 
ment contains neither log. of A nor a negative power of h, but it contains a frac- 
tional power of h, the exponent being between 1 and 2, which is not the true con- 
clusion. 



1HE DIFFERENTIAL CALCULUS. 107 

'■% — 6, then will the development hold when a + h is substituted for 
#• in Fx, provided h be taken between the limits h==0 and h = b — a, 
or more strictly, provided it does not exceed these limits. In like 
manner, if the development hold for all values of a: from x = b down 
to x = a, but fails for x = a, then will the development hold when 
b — /lis substituted for x for all values of h from h — 0, to h = 6 — a, 
t>ut it will not hold for the value of h immediately succeeding this 
last ; and it is obvious that h will always be subject to such restrictions 
unless the development holds, not merely for x — a, but universally. 
When, therefore, we find that for x = a the differential coefficients 
do not any of them become infinite, all that we can conclude is that 
the development of F (a it h) is according to Taylor's theorem for 
all values of h between some certain finite value K ; , which may in- 
deed be indefinitely small, and 0, and it is only when this is not the 
case that the theorem is said by analysts to fail. We have thought 
it necessary to point out these circumstances to me student, seeing 
that some authors, from not attending to them, have fallen into very 
important errors, and have laid down erroneous doctrines with respect 
to the failing cases of Taylor's theorem. Thus Mr. Jephson at page 
191, vol. i. of his Fluxionial Calculus, a work containing much valua- 
ble information, says * c It may further be observed that Taylor's theo- 
rem always fails when the assigned value of x causes any of the terms 
to become imaginary, and that this may take place without causino- 

the function itself to be imaginary ; thus take /a? = c + x s Vx ~a 

if we suppose x = O.fx = c.f'x = 0, but f'x,f"'x , ... all con- 
tain \/ — a." From this it would appear that Taylor's theorem 
may fail to give the true development in other cases besides those 
which cause the differential coefficients to become infinite, which, 
however, is not true. Whenever, for any particular value of x, Tay- 
lor's coefficients become imaginary, we must infer, agreeably to the 
statement in (4), that the function F(s + h) becomes imaginary for 
that value of x; h being of course limited as above explained. In 
the example just quoted, where 



f x — 2x Vx — a + 



Vx — a 

2x ix 3 



f'x == 2 y/x — a + 
&c. &c. 



108 THE DIFFERENTIAL CALCULUS. 

the function f(x+ h) becomes, when x = 0,/(0 + h) = c + U J 
y/h — a and the development is 



c + h 2 y/h — a = c -f Qh -f \/ — a /* 2 -f ■ /i 3 + &c. 

2v — a 

and this is the true development, for h must not exceed the limits 
h = 0, and h = a, since x = a causes the differential coefficients ta 
become infinite, and therefore the development to fail. 

With regard to the failing cases of Maclaurin's theorem, it may be 
observed that they are very different from the failing cases of Taylor's. 
Whatever be the form of the proposed function, its general develop- 
ment, according to Taylor's theorem, never fails ; but the failure of 
Maclaurin's theorem always arises from the form of the proposed 
function and it is the general development that fails, and consequently 
all the particular cases. For it is obvious that every function or any 
of its differential coefficients which become infinite when x = 0, will 
fail to be developable by Maclaurin's theorem. 

Before terminating these remarks it may be proper to observe that 
the student is not to attribute what analysts have been pleased to term 
the failing cases of these theorems to any defect in the theorems 
themselves ; on the contrary they would be very defective if they did 
not exhibit such cases. All that is meant is, that the function in par- 
ticular stales may fail to be developable according to Taylor's series,, 
and under particular forms it may fail to be developable according to 
Maclaurin's series ; so that, in fact, these theorems fail to give the 
true development only when that development is impossible. 

(74.) Let us now examine implicit functions, and let us suppose 
that x = a causes a radical to vanish from F# in consequence of a 
factor of it vanishing ; we have seen (68) that such radical will reap- 
pear in some of the differential coefficients, suppose it appears in the 
first which requires that the factor spoken of be x — a, then for x = a 
this coefficient will have more values than the proposed function, as 
it contains a radical more. But if the function that Fx or y is of x 
be only implicitly given, that is by means of an equation without radi- 

dy 
cals, we know (52) that the expressionfor -~- will be also without radi- 

(XX 

cals, and from such an expression it does not at first seem clear how 



THE DIFFERENTIAL CALCULUS. 109 

we are to deduce the multiple values alluded to, and which might be 
obtained by solving the equation for y and thus introducing the radi- 
cal. But since the expression for -y- appears under the form of a 

(XX 

fraction, viz. (52) 

du 
dy dx 

Tx = ~ djj ■ • ' * t 1 )' 

dy 
we readily perceive that one case is possible, and only one, in which 
this fraction may take a multiplicity of values besides those implied hi 
y, viz. the case in which it becomes £ ; that is, when the following 
conditions exist simultaneously, viz. 

du da 

-j- = 0, — = . . . . (2 ; 
dx ay 

so that these conditions are those which must exist for every value of 

x which destroys a radical in Fx but not in —. 

dx 

(75.) Whenever, therefore, any particular value of x destroys a 

du 
radical in y, but not in ~, then the expression (1) must take the form 

£, and admit of the necessary multiple values. 

The rules laid down in (41) enable us to determine the true value 
of the fraction (1 ) in the proposed circumstances, that is when it takes 
the form | , for there is but one independent variable, viz. x. By dif- 
ferentiating numerator and denominator separately, we have, by the 
article referred to, 

d 2 u d 2 u dy 



hence, 



r dy dx 2 dx dy dx 

^ = "~t"^7~ ■ tf u X] .... (3); 

+ . — - 

dx dy dif dx 



r dhl . 2 _^_ % , ^L dkm - ft 
W T dxdy ' dx ^ dy 2 W J 



This being a quadratic equation furnishes two values for [— 1, 



dx- 



110 THE DIFFERENTIAL CALCULUS. 

dv 
which are all that belong to the fraction (1) or to [-p] unless, indeed, 

both numerator and denominator of (3) alsovanishfor a value of xgiven 

by the conditions (2), in which case we must differentiate again as in 

dy 
art. (41 ), when the value of [~r~\ will be given by an equation of the 

third degree which will furnish three values, and so on ; and in gene- 

dy 
ral, if the fraction or \_~r\ admit of n values, the equation which de- 
termines them can be obtained only by differentiating?? times, which 
will lead to an equation of the ??th degree, aod it is plain that the ra- 

1 
dical destroyed in Fx must be of the -th degree, seeing that it gives 

dy 
to -j- n additional values. 
ax 

Suppose, for example, we had the function 

y = x + (x — a) \/ x — b 
dy 



± = l + Vx*-b + 



" dx 2Vx — b 

and for x = a 

[!/] =«>[£] = 1 ± V«-6 

. dy 
so that the radical which disappears in y appears in — ; this, there- 
fore, has two values. 

Now let the same function be given in an implicit form and with- 
out radicals, viz. 

U =(y-L- xf —{x — a) 2 (x — b) = 0, 

... ^=:__2 (y — x ) — (x — a) (Sx — 2b + a), -^ = 2(y — x). 
Since for x = a, y = ft, therefore both these expressions become 

: hence l"-^!"!-* Taking, then, the differentials of both numera- 
L dj7 J 

tor and denominator of the fraction, 

2 (y — x ) -f (x — a) (3x — 26 + ft) _ 

2(y — *) 0' 



THE DIFFERENTIAL CALCULUS. Ill 



we find when x = a that it becomes 



[|] = i+ a ~ b 



dx A_i 

l dx j 



therefore, 



[I- 1 ! 2 - ( ffi - 6 > =0.-. [J] =1 ± V«_6, 

as before. 

(76.) Suppose now that the radical which disappears from Fx by- 
reason of a factor, disappears also from — , and that it appears in —4, 

ax cix^ 

which is the same as supposing that the vanishing factor is (x — a) 2 . 

dv 
In this case -— will have the same number of values that Fx has for 
ax 

72 

x = a, but additional values will belong to -T^sothatwemusthave 

dhj _ 
tdx 2 J ~~ 

and the true values, when the function is implicit, will all be deter- 
mined by the principles already employed. 
For example : the explicit function 

y = x + (x — a) 2 V x 
gives when x = a 

and we shall see that the same values are equally given by the impli- 
cit function 

(y — x) 2 = (x — af x, 

by applying the foregoing process to -j-^-. 

For by successively differentiating, and representing, for brevity, 
the several coefficients derived from y by p\ p", &c. we have 

2 (p' — 1) (ff — *) = (* — a) 3 (5a? — 0) .-. [//] = 1 
(p' — l) 2 + p" (y — x) = 2 (x — a) 2 (5x — 2a) 



112 THE DIFFERENTIAL CALCULUS. 

* " U J L y — x J 

_ 6 {x — a) (5ar — 3a) — 2p" (p' — 1) _ 
- L p' — i J "" 

_ 60z — 48a — 2p'" (y f — 1) — 2p" 2 _ 12a — 2 [p" 2 ] 

- L p- —J = -^ 

... 3 [p"2] — 12a .-. [p"] ^ ± 2 v/ a, 

as before. 

The two examples following will suffice to exercise the student in 
this doctrine, which is merely an extension of the principles treated 
in Chapter V. to implicit functions. 

1. Given 

y 3 = (x — a) 3 (x — r b) 

to determine the values of — , when x = a, 
ax 

dij., 

tei = V— *• 

2. Given 

( y __ x f — {x — a) 4 (x — b)=0 

dy $y 

to determine the values of -g and -tV, when a; = a. 



We here terminate the First Section, having fully considered the 
various particulars relating to the differentiation of functions in gene- 
ral. 



THE DIFFERENTIAL CALCULUS. 



SECTION II. 



113 



APPLICATION OF THE DIFFERENTIAL CALCULUS 
TO THE THEORY OF PLANE CURYES. 



CHAPTER I. 
ON THE METHOD OF TANGENTS. 

(77.) We now proceed to apply the Calculus to Geometry, and 
shall first explain the method of drawing tangents to curves. 

The general equation of a secant passing through two points (#', 
y f ), [x'\ 7/"), in any plane curve, is {Anal. Gccr.i.) 

v' — y' 

y' — y'\ being the increment of the ordinate or proposed function 
corresponding to x — x" the increment of the abscissa or independ- 
ent variable. The limit of the ratio of these increments, by the prin- 

ciples of the calculus, is --,-, ; that is to say, such is the representation 

of the ratio when x — x" = 0, and, consequently, y' — y" = 0. 
But when this is the case the secant becomes a tangent. Hence the 
equation of the tangent, through any point (a?', y') of a plane curve, is 

df 

y--y' := fa'( x ~- af ) • - - ° C 1 )-. 

di,' 

It appears, therefore, that the differential coefficient -r-, for any 

proposed point in the curve has for its value the trigonometrical tan- 
gent of the angle included by the linear tangent and the axis of x, that 
is, provided the axes are rectangular. If the axes are oblique, the 
same coefficient represents the ratio of the sines of the inclinationo 
of the linear tangent to these axes. (See Anal. Geom.) 

By means of the general equation (1) we can always" readily de- 
termine the equation of the tangent to any proposed plane curve when 
the equation of the curve is given, nothing more being required than 
to determine from that equation the differential coefficient. 

15 



114 THE DIFFERENTIAL CALCULUS. 



Suppose, for example, it were required to find the particular fonr/ 

jre have to determine - 

< 

Aif' a +.B*' a = A 2 B 2 t 



dy' 
for the ellipse. We here have to determine — J — from the equation 



and which is 

dy' _ BV 

dx' A 2 y' 

therefore the equation of the tangent is 

y — y = — ^y (* — *)• 

(x, y) being any point in the curve, and (x, y) any point in the tan- 
gent. 

Again ; let it be required to determine the general equation of the 
tangent to a line of the second order. 

By differentiating the general equation 

Ay' 2 + Bx'y' + Cx' 2 + By' + Ear' + F = 0, * 
we have 

^g+B'g + Bjr' + lW + Dg +E = o r 

dy' _ 2CV + By' + E 

""' ~dx~' 2Ay' + Bx' +~D 

so that the general equation is 

2(V + By' + E 

(78.) As the normal is always perpendicular to the tangent, its 
general equation must be, from (l) r 

y-y' = -^(- x -^ t — oo- 

eta' 

* The general equation of lines of the second order in its most commodious 
form is 

from which, by differentiation, we have 

dy' m-f- 2nx' 

lx'~ 2y~' 
and the equation of the tangent to a line of the second order is therefore 

. m -f 2nx' 

y-y= 2y , (*-*'). 

dx' 
f K — — (a — *'). 

dy' Ed. 



THE DIFFERENTIAL CALCULUS. 115 

tt is easy now to deduce the expressions for the subtangent and sub- 
normal. For if, in the equation of the tangent, we put y = 0, the 
resulting expression for x — x' will be the analytical value of that part 
of the axis of x intercepted between the tangent and the ordinate y' of 
the point of contact, that is to say, it will be the value of the subtan- 
gent T, (Anal Geonu),* 

.-. T = — ?- .... (3% 

dx' 
If, instead of the equation of the tangent, we put y = in that of the 
normal, then the resulting expression for x — x' will be the value of 
the intercept between the normal and the ordinate y\ that is, it will 
belong to the subnormal N p 



As to the length of the tangent T, since T = Vy* + T, 2 , we 
3have, in virtue of (3), 

T=yy(i + ^)....(5). 

dx' 2 



Also, since the length of the normal N is N = sf y' z -f N /a we 
have, by (4), 

• N = y ' v (i + 2& — (6) - 

The foregoing expressions evidently apply to any plane curve what- 
ever, that is, to any curve that may be represented by an equation 
between two variables, whatever be its degree, or however compli- 
cated its form. 

We shall now give a few examples principally illustrative of the 
method of drawing tangents to the higher curves, for which purpose 
we shall obviously require only the formula (3), for it is plain that to 
any point in a curve we may at once draw a tangent, when the length 
and position of the subtangent is determined. Or, knowing the point 
(a/, J/'), we may, by putting successively x — and y = 0, in the equa- 
tion (1), determine the two points in which the required tangent ought 

dv' 
to cut the axes of the coordinates and then draw it through them. If ~- 

dxr 



1 16 THE DIFFERENTIAL CALCULUS. 

is at the proposed point, the tangent will be parallel to the axes of x, 

du' 
because, as remarked above, — - is the value of the trigonometrical 

dx 

tangent of the inclination of the linear tangent to the axes of x, and for 

. , dy' 
this reason also the tangent will be parallel to the axes oiy when — - r 

i3 infinite at the proposed point. 

EXAMPLES. 

(79.) 1. To draw a tangent to a given point P in the common 
or conical parabola. 

By the equation of the curve 




f 


= px 


d£ 


__ P 


dx' 


" 2y 



.-. T, = y' ~ JL = ?C = 2x. 

Hence, having drawn from P, the perpendicular ordinate PM, if we 
set off the length, MR, on the axis of x, equal to twice AM, and then 
draw the line RP, it will be the tangent required. 

2. To determine the subtangent and subnormal at a given 
point (x' 9 y r ) in the parabola of the nth. order, represented by the 
equation 

y = ax" 
dy' , , 

dx' 

.*. T = y -7- nax" 1 - 1 = -, N, = y' nax n ~ x == na 2 x 2n ~ l or —-. 

J n J x 

3. To determine the subtangent at a given point in the loga- 
rithmic curve. 

The equation of this curve related to rectangular coordinates is 

y = a*, 
which shows that if the abscissas x be taken in arithmetical progres- 
sion, the corresponding ordinates y will be in geometrical progres- 
sion, so that the ordinates of this curve will represent the numbers, 
the logarithms of which are represented by the corresponding ab- 



THE DIFFERENTIAL CALCULUS. 117 

scissas, a being the assumed base of the system. Hence, calling the 
modulus of this base hi, we have, by differentiating (13), 

dy _ 1 

Tx~m Vi 

.-. T, = y + &- = m. 

' u m 

Hence the remarkable properly that the subtangent is constantly equal 
to the modulus of the system, whose base is a. 

4. To determine the subtangent at a given point in the curve 
whose equation is 

x 3 — 3axy + y 3 = 0. 



Here 



A =3*2— Zay — 3«^'+ Zy 2 ^- = 0, 
dx s dx J dx 

dy' _ ay' — x' 2 

•** 17 ~ T» 

dx y — ax 



ay — x" 

5. To draw a tangent to a rectangular hyperbola between the 
asymptotes, its equation being xy = a. 

T = x. 

6. To determine the subtangent at a given point in a cuiwe 
whose equation is y m = ax n , which, because it includes the common 
parabola, is said to represent parabolas of all orders. 

n 

7. To determine the subtangent at a given point in a curve 
whose equation is x m y n = a, which, because it includes the com- 
mon hyperbola, is said to belong to hyperbolas of all orders. 

m 
(80.) If the proposed curve be related to polar coordinates, then 
the expressions in last article must be changed into functions of these. 

If the curve AP be related to polar coor- .. 

dinates FP = r, PFX = w, then if PR be -^~~ 

a tangent at any point, P and PN the nor- ix^/ / j \ 

mal, and if RFN be perpendicular to the ra- a - j^;^ \ x 

dius vector FP, the part FR will be the polar \\ 

subtangent, and the part FN the polar sub- 



118 THE DIFFERENTIAL CALCULUS. 

normal. When the pole F and the point P are given, it is obvious 
that the determination of the subtangent FR will enable us to draw 
the tangent PR. 

The formulas for transforming the equation of a curve from rec- 
tangular to polar coordinates, having the same origin, are (Anal. 
Geom.) 

x = r cos oj, y = r sin. w, x 2 + y 2 = r 2 , 
and the resulting equation of the curve will have the form 

r = Fw, or F (r, w) = 0, 
in which we shall consider w as the independent variable. Now 
RF = PF • tan. /P=r tan. /_ P, but by trigonometry, 
. _ tan. w — tan. a 





1 -\- tan. oj tan. a 


that is, since 




' 


dy . y 

tan. a = ~r- and tan. w = - 

ax x 




y dy 

tan. Z P= X f 

x ax 



therefore r times this expression is the value of the polar subtangent. 

dii 
But the differential coefficient -p, which implies that x is the princi- 



pal variable, 


ought to become, when the variable is 


changed to w, 


(66), 


dx 


dy_ ^ 
du 


— — .-. tan. / P = 
du 


dx 


dy_ 

eta 

dif 












x . _i_ 

eta ^ 


yiur 


Also, f 


rom the above formulas 


of transformation, 




dx 
du 


_ dr 

du 


- COS. 60 


— r sin. 


eta 


dr . , 
— — sin. 6J + 

CtOO 


r cos. 60 




•*• 


dx 


_ dr 

c/w 


sin. co cos 


. 60 -- r 2 sin. 


2 60 






dy 

x — — 
eta 


eta 


sin. w cos 


. 60 + r 2 cos 


2 W 



THE DIFFERENTIAL CALCULUS. 119 





dx dr _ „ . 
x — — = r -7— cos. co — r 4 sin. w cos. w 
dw aw 




w — p- = r — — sin. u + ir sin. w cos. w 
dw dw 


whenc 


9 

dx dy _ d# . du dr 
2/ -j a? -5^- = — r, a? -j- + 2/ -j*- = r - T - 

J dw dw dw dw dw 


consec 


uently, 




r r 2 
tan. / P = -7- .'. RF = — j— = subtangent. 
dr dr 




dw dw 


Also 






FP 2 r 2 dr 
FN = t— — = r 2 — —r- = -r— = subnormal. 
Fit dr dw 




dw 

(81.) We shall apply these formulas to spirals, a class of curves 

always best represented by polar equations. 

8. To determine the subtangent at any point in 

the Logarithmic Spiral, its equation being 

w 
r = a 

dr . w 

.\ — : — = log. a a = 

dw m 

2 • dr rr 

.*. r J — -= — = mr = I,. 

dw 

Hence, if a represent the base of the Napierian system, since the 
modulus will be 1, the subtangent will be equal to the radius vector, 
and therefore the angle P equal to 45°, because tan. /_ P = 1. 

Since, by the equation of this curve, log. r = w log. a, it follows 
that, if a denote the base of any system, the various values of the 
angle or circular arc w will denote the logarithms of the numbers 
represented by the corresponding values of r. Hence, the propriety 
of the name logarithmic spiral. In this curve 

tan. / r = r — —f- = a — = m : 

dw m 

hence the curve cuts all its radii vectores under the same angle. 



120 THE DIFFERENTIAL CALCULUS. 

9. To determine the subtangent at any point in the Spiral of 
Archimedes, its equation being 

r — aw 

dr 2 • dr > rr 

.*. — — = a .\ r 2 — -j— = au A = rw = T. 
aw aw 

so that FR is equal to the length of the circular arc to radius r t com- 
prehended between FR, FA ; when, therefore, w = 2tf, the subtan- 
gent equals the length of the whole circumference. The spiral of 
Archimedes belongs to the class of spirals represented by the general 
equation 

r = aw". 
When n = — 1, we have ru = a, and the spiral represented is 
called the hyperbolic spiral, on account of the analogy between this 
equation and xy = a. It is also called the reciprocal spiral. 

10. To determine the polar subtangent at any point in the hy- 
perbolic spiral. 

T, = a. 

11. To determine the polar subtangent at any point in the spiral 

_x 
whose equation is r = aw 2 . 

1 In 2 
T = 2aw~ 2 = — . 
r 

12. To determine the polar subtangent at any point in the para- 
bolic spiral, its equation being r = aw 2 . 

T =?!l 

a 2 ' 

13. To determine the polar subtangent at any point in a spiral 
whose equation is 

( r 2 _ „,.) w 3 _* I = 0. 

_ i(a — 2r)r» 

( r 2_ ar )2 

Reftilinear Asymptotes. 

(82.) ^2 rectilinear asymptote to a curve may be regarded as a 
tangent of which the point of contact is infinitely distant, so that the 
determination of the asymptote reduces to the determination of the 



THE DIFFERENTIAL CALCULUS. 121 

tangent on the hypothesis that either or both y' — 0, x' = 0, the 
portions of the axes between the origin and this tangent being, at the 
same time, one or both finite. 

The equation of the tangent being 

ive have, by making successively y = 0, x — 0, the following ex- 
pressions for the parts of the axes of x and y, between the tangent 
raid the origin, viz. 

*'_i an d^_*'-!C .... (i). 

dif J dx' K ' 

w 

If for x = oo both these are finite, they will determine two points, 
one on each axis, through which an asymptote passes If for x = oo 
the first expression is finite and the second infinite, the first will de- 
termine a point on the axis of x, and the second will show that a line 
through this point and parallel to the axis of y is an asymptote. If, 
on the contrary, the second expression is finite, and the first infinite, 
the asymptote will pass through the point in the axis of i/, determined 
by the finite value, and will be parallel to the axis of x. 

When, however, asymptotes parallel to the axes exist, they may 
generally be detected by merely inspecting the equation, as it is only 
requisite to ascertain for what values of x, y becomes infinite, or for 
what values of?/, x becomes infinite. Thus, in the equation a??/ = a, 
x = 0, renders y = oo , and y = renders x = go , therefore the 
two axes are asymptotes. Again, in the equation 

bx 4 
a" y 2 — y 2 x 2 — bx d ' = 0, or y 2 = — • 

J * J a 2 — x 2 

it is plain that x = ± a renders y = oo, we infer, therefore, that the 
curve represented by this equation has two asymptotes, each parallel 
to the axis of y, and at the distance a from it. 

If both expressions are infinite, there will be no asymptote corres- 
ponding to X = 00 . 
If both expressions are 0, the asymptote will pass through the origin, 

and its inclination d, to the axis o(x will be determine, d by -f— = 

J dx' 
tan. L 

16 



122 THE DIFFERENTIAL CALCULUS. 

If for y = qo one or both of the above expressions are finite, there 
will be an asymptote, and its position may be determined as in the 
foregoing cases. 

EXAMPLES. 

(83.) 1. Let the curve be the common hyperbola, of which the 

equation is 

b 

y = -Vx 2 —a 2 

dy _ bx 

' dx a x /^_ a 2 

hence the general expressions (82) are 



and 

b a 2 ba 



a j x 2 — a 2 J a 2 

* l ~* 

both of which are 0, when x — oo ; hence an asymptote passes through 
the origin. 
Also 

dy b 1 



dx a j a 2 

which becomes ± — , when x = oo , therefore, this being the tangent 
a 

of the inclination of the asymptotes to the axis of x, they are both rep- 
resented by the equation 

. b 
y = ± - x. 

2. To prove that the hyperbola is the only curve of the second 
order that has asymptotes. 

The general equation of a line of the second order, when referred 
to the principal diameter and tangent through the vertex as axes, is 
w 3 = mx + nx 2 f 



THE DIFFERENTIAL CALCULUS. 



123 



y _ 2y 2 _mx + 2nx 2 — 2if _ mx 

X ~"dy~ X ~~m J r 2nx~ m + 2nx m + 2nx 

dx 

dy '_ mx + 2az 2 _ 2y 2 — mx — 2nx 2 _ mx 

V ~~ X dx~~ y <ty~ ~2y~~ 2 Vmx~+nx 2 ' 

Dividing numerator and denominator of each of these expressions by 
x, they reduce to 

m 



and 



w» . _ I 



Hi 



X 



2" 2 V~ + n 



x 



and these, when x = go , or indeed when y = co , become 



m m 

and 



2n 2Vn 

Hence the curve will have asymptotes, provided n be neither nor 
negative, that is to say, provided the curve be neither a parabola nor 
an ellipse, but if it be either of these, there can exist no asymptote ; 
therefore the hyperbola is the only line of the second order which has 
asymptotes. 

(84.) When the curve is referred to polar coordinates, then, since 
the radius vector of the point of contact is infinite when the tangent 
becomes an asymptote, it follows that if for r = go the subtangent 
is finite, this subtangent may be determined by (80) in terms of w, 
and w may be found from the equation of the curve, so that there 
will thus be determined a point in the asymptote and its direction, 
which is all that is necessary to fix its position. There will always 
be an asymptote if w is finite, for r = co . If, for r = go , w is also 
go , there exists no asymptote. 

3. Let the curve be the hyperbolic spiral. 

By ex. 10, art. 81 , the subtangent at any point is constant, and equal 
to a, therefore there must be an asymptote ; also 



a 



by the equation of the curve w — - = 0, when 

r 

r = go , therefore the asymptote is perpendicular 

to the fixed axis at the distance a from the pole. 

Neither the logarithmic spiral, nor the spiral of 

Archimedes have an asymptote. 




124 THE DIFFERENTIAL CALCULUS. 

4. Let the spiral whose equation is 



^ A — 1 1 — w~ J 

be proposed, which admits of a rectilinear asymptote, because eo = 1 
renders r = go . The direction, therefore, of the asymptote is ascer- 
tained, and consequently the direction of the infinite radius vector, 
since they must be parallel. It remains, therefore, to determine the 
subtangent, or distance of the asymptote from the pole 

dr _ 2aur 3 2r 2 

c/w (1 — w -2 ) 2 ' aw 3 

r 2 — — = 2 — -?!f — f!f. — - = T 

doo V wJ " 2 ~ 2 ~ ' 

because w = 1 when r = go . 

(85.) Although we do not propose to treat fully in this place of 
curvilinear asymptotes, yet we may remark in passing, that if r should 
be finite although gj be infinite, it will prove that the spiral must be 
continued for an infinite number of revolutions round its pole, before 
it can meet the circumference of a circle whose radius is this finite 
value. In such a case, therefore, the spiral has a circular asymptote. 
If, moreover, the value of r for w = go be greater than the value of r 
for every other value of w, the spiral will be included within its circu- 
lar asymptote, but, otherwise, it will be without this circle. 
5. Thus in the spiral whose equation is 

(r 2 — ar) or — 1 = or w = — 

>/ r 2 — ar 

u is infinite when r = a, and for all less values of r, w is imaginary ; 

hence the spiral can never approach so near to the pole as r =■ a, till 

after an infinite number of revolutions, so that the circumference 

whose radius is a is within the spiral and is asymptotic. 

If, on the contrary, the equation had been 

1 



\/ ar — r 2 
then also r = a gives u = go , but for all other real values of w, r is 
less than a, so that this spiral is enclosed by its asymptotic circle, the 
radius of which is a. 



THE DIFFERENTIAL CALCULUS. 125 

6* To determine the rectilinear asymptote to the logarithmic 
curve. 

The axis of x. 

7. To determine the equation of the asymptote to the curve 
whose equation is 

y 3 =ax 2 + x 3 . 

The equation is y = x -j- £ a. 

8. To determine the rectilinear asymptote to the spiral whose 
equation is 



= au 2 



The fixed axis is the asymptote. 
9. To determine whether the spiral shown to have a rectilinear 
asymptote in ex. 4 has also a circular asymptote. 

The circle whose centre is the pole and radius = a is an asymp- 
tote. 

(86.) Before terminating the present chapter, it will be necessary 
to exhibit the expression for the differential of the arc of any plane 
curve, as we shall have occasion to employ this expression in the next 
chapter. 

Let us call the arc AB of any plane curve s, and 
the coordinates of B, x, y ; let also BD be a tan- 
gent at B, and BC any chord, then if BE, ED are 
parallel to the rectangular axes, BC will be the in- 
crement of the arc s corresponding to BE — h, the 
increment of the abscissa x. I 

Now, putting tan. DBE = a, we have 



ED =h*.'. BD = V h 2 



and 



BD+DC = V h 2 {\ + a 2 ) + ha— CE 
BC V h 2 + CE 2 

CE 

This ratio continually approaches to -= or to unity as h diminish- 

CE 

es and this it actually becomes when h = 0. Consequently, since the 

arc BC is always, when of any definite length, longer than the chord 



126 THE DIFFERENTIAL CALCULUS. 

BC and shorter than BD + BC * it follows that when h = that 
the ratio of the arc to either of these must be unity ; therefore 

. , _ . arc BC , arc BC chord BC 

in the limit —. z-^r^ = 1 •*• ; -r- i = 1, 

chord BC h h 

but 



chord BC V h 2 + CEj = V CW 



h h 

and CE is the increment of the ordinate y corresponding to the incre- 
ment h of the abscissa x ; hence, when h = 0, the ratio becomes 



ds _^ V 1 dy 2 
dx dx 2 



d* = V df_ 

''' dx~ l + dx 2 ' 
If any other independent variable be taken instead of x, then, denoting 
the several differential coefficients relatively to this new variable by 

(dx), (dy), (ds) we have (66) 



(&)= V(dxy + (dyf. 
At the point where — - = 0,— = 1, or (ds) = (dx), 



CHAPTER II. 



ON OSCULATION, AND THE RADIUS OF CURVATURE 
OF PLANE CURVES. 

(87.) Let 

y=fx,Y = Fx, 

be the equations of two plane curves, in the former of which we shall 
suppose the constants a, b, c, &c. to be known, and therefore the 
curve itself to be determinate ; while in the latter we shall consider 
the constants A, B, C, &c. to be unknown, or arbitrary, and there- 

* See Young's Elements of Plane Geometry. 



THE DIFFERENTIAL CALCULUS. 127 

fore the species only of the curve given. The constants which enter 
into the equation of a curve, are usually called the parameters. 

If, now, x take the increment h, and the corresponding ordinates 
y', Y' be developed, we shall have, by Taylor's theorem, 
, . dy . , d 2 y h 2 d?y h 3 , .... 

T ,_ y4 .<flf,,fl * , <FY ft 3 , „ ,.' 

T - Y + i 4 %R + tfRl + &c (2) - 

Now, the parameters which enter (2) being arbitrary, they maybe 
determined so as to fulfil as many of the conditions 

^ dy dX d 2 y d 2 Y 

as there are parameters, but obviously not more conditions. 

We shall thus have the values of A, B, C, &c. in terms of x, and 
of the fixed parameters a, 6, c, &c. ; which values, substituted in (2), 
will cause so many of the leading terms in both series to become 
identical, whatever be the value of a?. Other corresponding terms of 
the two series may, indeed, be rendered also identical, but this can 
take place only accidentally, not necessarily. Hence, whatever par- 
ticular value we now give to x, the resulting values of the corres- 
ponding coefficients will necessarily agree to the extent mentioned, 
that is, as far as the n first terms, if there are n constants originally 
in (2) ; and this is true, even if such particular value of x render any 
of the coefficients infinite, inasmuch as they are always identical as 
far as these terms, but no further. 

We know, however, that in those cases where any of the coeffi- 
cients become infinite, (.1) and (2) will fail to represent the true de- 
velopments of the ordinates y\ Y' at the proposed points. Neverthe- 
less, as the two series have been rendered identical, as far as n terms, 
should they both fail within this extent, the terms which supply these 
in the true development, must necessarily be identical. [See note C 
at the end. ) 

Now the greater number of leading terms in the two developments, 
which are identical, the nearer will the developments themselves ap- 
proach to identity, provided, at least, h may be taken as small as we 
please ; for if n — 1 terms in each are identical, we may represent 
the difference of the two developments by 



128 THE DIFFERENTIAL CALCULUS. 

A n h a + S — (A' n h*' -f S') (4), 

where S, S' represent the sums of the remaining terms in each series 

after the nth. Hence, h a being the highest power of h which enters 
this expression, for the difference it follows from (47), that a value 

may be given to h small enough to cause the term A„ h a to become 
greater than all the other terms in (4), and consequently, for this 
small value, 

A n h a — A! n h*' 7S-S', 
and, therefore, the whole difference (4) is greater than twice S — S', 
but when the nth term is the same in both developments, as well as 
the preceding terms, then the difference (4) is reduced simply to 
S — S', which we have just seen to be less than (4). Consequently 
the developments approach nearer to identity, for all values of h be- 
tween some certain finite value hi and as the number of identical 
leading terms become greater. 

When the first of the conditions (3) exist, the curves have a com- 
mon point ; when the second also exists they have a common tangent 
at that point, and are consequently in contact there, and the contact 
will be the more intimate, or the curves will be the closer in the 
vicinity cf the point, as the number of following conditions become 
greater ; so that of all curves of a given species, that will touch any 
fixed curve at a proposed point with the closest contact whose para- 
meters are all determined agreeably to the conditions (3). No other 
curve of the same species can, from what is proved above, approach 
so nearly to coincidence with the proposed, in the immediate vicinity 
of the point of contact, as this ; so that no other of that species can 
pass between this and the proposed. A curve, thus determined, is 
said to be, in reference to the proposed curve, its osculating curve of 
the given species. 

(88.) It appears, from what has now been said, that there may be 
different orders of contact at any proposed point. The two first of 
the conditions (3) must exist for there to be contact at all ; therefore, 
when only these exist, the contact is called simple contact, or contact 
of the first order ; if the next condition also exist, the contact is of 
the second order, and so on; and it is obvious, that of any given 
species, the osculating curve will have the highest order of contact, 



THE DIFFERENTIAL CA1XULUS. 129 

at any proposed point, in a given curve. If the curve, given in 
species, has n parameters, the highest order of contact will be the 
n — 1th, unless, indeed, the same values of these parameters that 
fulfil the n conditions (3), should happen also to fulfil the n + 1th, 
the n + 2th, &e. ; but this, as observed before, can take place only 
accidentally, and cannot be predicted of any proposed point, although 
we see it is possible for such points to exist. 

(89.) At those points in the proposed curve, for which Taylor's 
development does not fail, contact of an even order is both contact 
and intersection, and contact of an odd order is without intersection ; 
before proving this, however, we may hint to the student that contact 
is not opposed to intersection, for two curves are said to be in con- 
tact at a point, when they have a common tangent at that point; and 
yet, as we are about to show, one of these curves may pass between 
the tangent and the other, and so intersect where they are admitted to 
be in contact. To prove the proposition, let us take the difference 
(4), which, when Taylor's theorem holds, is 

(A n -A' n )h a + S-s' (5,) 

A„ A' n being here the n — 1th differential coefficients. If these are 

odd, the contact is of an even order, also a being odd, h a will have 
contrary signs for h = -f h' and h = — h', and therefore, since for 
these small values of h, the sign of the whole expression (5) is the 
same as that of the first term, the differences of 
the ordinates corresponding to x + h, and to 
x — h, will be the one positive and the other 
negative, so that the two curves must necessarily 
cross at the point whose abscissa is x. 

But if a is even, the contact is of an odd order, and the difference 
(5) between the ordinates of the two curves corresponding to the 
same abscissa, x + h, will, for a small value of h, have the same 
sign, whether h be positive or negative ; so that, in this case, the 
curves do not cross each other at the point of contact. 

(90.) The student must not fail to bear in remembrance, that the 
proposition just established, comprehends only those points of the 
proposed curve, at which none of the differential coefficients become 
infinite from the first to that immediately beyond the coefficient which 
fixes the order of the contact, For it is only upon the supposition 

17 




130 THE DIFFERENTIAL CALCULUS. 

that the true development, within the limits, proceeds according to 
the ascending integral and positive powers of A, that the forego : ng 
conclusions respecting the signs of the difference (5) can be fairly 
drawn. (See note C.) 

(91.) From the principles of osculation now established, it is evi- 
dent that any plane curve being given, and any point in it phosen, we 
may always find what particular curve, of any proposed species, shall 
touch at that point with the closest contact, or which shall most 
nearly coincide with the given curve in the immediate vicinity of the 
proposed point. Thus an ellipse or a parabola being given, and a 
point in it proposed, we may determine the circle that shall approach 
more nearly to coincidence with that ellipse or parabola in the vicinity 
of the proposed point, than any other circle, and which will therefore 
better represent the curvature of the given curve at the proposed point 
than any o her. On account of its simplicity and uniformity, the 
circle is the curve employed to estimate, in this way, the curvature of 
other curves at proposed points; that is, the curvature is estimated by 
the curvature of the osculating c'rde, or rather as the curvature of 
a circle increases as the radius diminishes, and vice versa, it is usual 
to adopt, as a fit representation of the curvature, the reciprocal of the 
radius. 

The osculating circle is called the circle of curvature, and its ra- 
dius the radius of curvature, and, from what has been said above, it 
follows that the determination of the curvature at any point in a pro- 
posed curve, reduces itself to the determination of this radius : to this, 
therefore, we shall now proceed. 

Radius of Curvature. 

I 

PROBLEM I. 

(92.) To determine the radius of curvature at any proposed point 
of a given curve. 

The general equation of a circle being 

(x — a) 2 + (y— /3) 2 = r 2 , 
it becomes determined as soon as we fix the values of the parameters 
a, (3, r, and these may be determined, so as to fulfil any three inde- 
pendent conditions, but not more. In the case before us, the condi- 
tions to be fulfilled are those of (3) art. (87), that is to say, putting 



THE DIFFERENTIAL CALCULUS. 131 

'p, p", &c. for the successive differential coefficients derived from Y 
= Fxf the equation of the given curve, the conditions to be fulfilled 
are 

V 1 >dx P, dx* P ' 
in order that the resulting values of a, /3, r, may belong to the equa- 
tion of the osculating circle. Now 

*% _ x — a dy 2 _ 1 (a? — a) 2 _ r 2 

hence the three equations for determining a, (3 and r, are 
(x-*y+ {y — i3) 2 =r*....(l), 
(*__«) +y(y_-0) =0.«. .(2), 

K(2/-/3) 3 = ~^....(3) 
From the second equation 

(*- a ) 2 =p' 2 (*/-/3) 2 . 
Substituting this in the first, 

(?' 2 + l)(?/-/3) 2 = r 3 . 
Adding this last to the third, there results 

which, substituted in (2), gives 

' s X - P(P' 2+ 1 ) . r2 - (p' 2 +l) 3 
J>" P'" 2 ' 

Consequently, 

P(p- + 1) j," + 1 

* p" ,iW J+ p" 

cfo 3 
(p'2 + l)f _ IhF 
r p" ' T 77 * 

These equations completely determine the osculating circle, when- 
ever -the co-ordinates x, ij of the proposed point are given. 

Should this point be such as to render p = 0, then the expression, 
for the radius of curvature at that point, becomes 



132 THE DIFFERENTIAL CALCULUS. 

-_ .1- T 

p" <jY 

dx 2 
But when p' = 0, the tangent at the proposed point must be pa- 
rallel to the axis of x (78), or, which is the same thing, the axis of y 
must coincide with the normal ; hence, under this arrangement of the 
axes, x = at the proposed point, and therefor© 
_ 1 

1 dx 2J 
Should p" = at the proposed point, r will be infinite, whether 
p' = o or not, so that the osculating circle then becomes a straight 
line ; as, therefore, this straight line has contact of the second order, 
/ the parts of the curve in the vicinity of the point 
^" will lie on contrary sides of it, as in the annexed 

diagram (89), that is, supposing p'" is neither nor 
oo . lip 1 " = 0, and the next following differential coefficient nei- 
ther nor co , the contact will be of an order which is unaccompanied 
by intersection. 

A point at which the tangent intersects the curve, or at which the 
curve changes from convex to concave, is called a point of inflexion, 
or, a point of contrary flexure. The analytical indications of such 
points will be more fully inquired into, when we come to speak of the 
singular points of curves. 

(93.) By referring to equation (2) above, which has place even 
when the contact is but of the first order, we learn that the centre 
(a, (3) of every touching circle, is always on the normal at the point 
of contact ; for that equation is the same as 

c—»> = -i <*-»>■ 

dx 
We shall now apply the general expression, for the radius of cur- 
vature, to a few particular cases. 

EXAMPLES. 

(94.) 1. To determine the radius of curvature, at any point in a 
parabola. 



THE DIFFERENTIAL CALCULUS. 133 

Differentiating the equation of the curve, 

y 2 = 4m#, 

we have, 

« , , 2m 

2yp = 4m .*. p = — 

2^ + 2^ = 0.-.^-^ = -^ 

(p'2+l)2 f m+x $ y 3 , |2 

p" K x 4m 2 ' m 2 

= - r-^— (See Anal. Geom.) 

Am 2 

As the expression for the normal diminishes with x, the vertex is 
the point of greatest curvature, r being there equal to 2m, or to half 
the parameter. 

2. To determine the radius of curvature at any point in an el- 
lipse. 

By differentiating the equation 

ay + 6V = a 2 b% 
we have 

a 2 yp' + b 2 x = .-. p' = ^ 

a 2 y 

a 2 yp" + a 2 p' 2 + b 2 = .-./>" = — *-il?*l = _ -|l 
Jr J J a J ?/ a 2 y 3 

_ (p ,2 + l) t _ (ay+6V)^ ay 3 _ (aV+feV)* 

From this expression, others occasionally useful may be readily 

derived. Thus, since (Anal. Geom.) the square of the normal, N, is 

6 4 

— x 2 + y 2 , therefore, 

a 4 3 

«<N 2 = 6V + ay ... r = -^- = ^ N 3 . . . . (2). 

Again, since (Anal. Geom.), 

b' 3 

aN = bb' .>.r = \ (3). 

ao 



134 THE DIFFERENTIAL CALCULUS. 

b 2 
At the vertex r=— = semiparameter (Anal. Geom.) 

From equations (2) and (3) it follows that, in the ellipse, the radius 
of curvature varies as the cube of the normal, or as the cube of the 
diameter parallel to the tangent through the proposed point. 

It is often desirable to obtain r as a function of X, the angle inclu- 
ded between the normal and the transverse axis. For this purpose 
we have since 

a* = a 2 (1 — it) and f = N 2 sin. 2 X 
a u 



'. N 2 Jl-(l--^)sin. 2 X^ = h 



a~ ' cr 



but {Anal. Geom.) 



N 



V 1 

a 



(1 — e 2 sin. 2 X)2 



... r = °L N 3 = b2 = a (1-V) 

a (1 — e 2 sin. 2 X) » ( 1 — e 2 sin. 2 X) 2 

(95.) Since, in the ellipse, the principal transverse is the longest 
diameter, and its conjugate the shortest (Anal. Geom.), it follows 

from (3), that the curvature*- is greatest at the vertex of the trans- 
verse, and least at the vertex of the conjugate axis. At the former 

& 2 , , , « 2 

point r = — - , and at the latter r = -7-. 
r a 4 b 

The present is a very important problem, being intimately con- 
nected with inquiries relative to the figure of the earth. 

By means of the last expression for r, the ratio of the polar and 
equatorial diameters of the earth, may be readily deduced, when we 
know the lengths of a degree of the meridian in two known latitudes, 
L, I, for these lengths may, without error, be considered to coincide 
with the osculating circles through their middle points ; and since 



THE DIFFERENTIAL CALCULUS. 135 

similar arcs of circles are as their radii, we have, by putting M, wi for 

the measured degrees, and R, r for the corresponding radii, 

R : r : : M : m, 

but 

a(l—e 2 ) , a (l—e 2 ) 
R = i '— - and r = J —, 

(1 — e 2 sin. 2 L)^ (1 — e 2 sin. 2 /) 2 

therefore, since mR = Mr, we have 

m M 



or 



(1— e 2 sin. 2 L) i (1 — e 2 sin. 2 /y 



, n 2 (l — e 2 sin>2 ^ = M 3 (! — e 2 sin. 2 L), 

2 , 6 2 M»- «»3 

.-. e 2 = 1 = 

a 2 2 2 



a_ M' sin. 2 L — m 3 sin. 2 / 
... _ y i __ _ 

m 3 cos. 2 / — M 3 cos. 2 L 



■'{ 



2. 
2 T / m3 N ' 2 7 

sin. 2 L — (-^-) sin. 2 / 

2 

(g[) cos. 2 / — cos. 2 L 



If/ = 0, that is, if the degree m is measured at the equator, then, 
a sin. L 

i^tei ~ ? * 

( M } C0S * L 

3. To determine the radius of curvature at any point in the loga- 
rithmic curve, its equation being y = a*, 

(m 4 + y 2 )i , . 
r = — , «i being the modulus. 

4, To determine the radius of curvature at any point in the cu- 
bical parabola, its equation being if = ax, 

&a?y " 



136 THE DIFFERENTIAL CALCULUS. 

PROBLEM II. 

(96.) To determine these points in a given curve, at which the 
osculating circle shall have contact of the third order. 

It is here required to find for what points of a given curve the val- 
ues of a, /3, r, determined by the three first conditions (3), art. (87) 
satisfy also the fourth condition. 

The differential coefficient p'" as derived from equation (3), p. 
131, is 

p y-P 1 

and this must agree with the p" derived from the equation of the pro- 
posed curve, at those points where the contact is of the third order ; 
that is, the abscissas of these points will all be given by the roots of 
the equation 

(y — P)p"' + 3p" P ' = o, 

and it may be easily shown, that the points which satisfy this equa- 
tion are those of greatest and least curvature, for since 





._ ( P ' 2 +i)* 




p" 


dr _ 


— 3(p'* + l)Vp" 2 + (P' 3 + 1)V 


dx 


p" 2 



and when r is a maximum or a minimum this expression is equal to 
(49) ; hence 

— 3 ? V /2 + Q/ 2 + l)p"' = 0, 

or, dividing by p" and recalling the value of y — (3 deduced in (92), 
we have, finally, 

(y-P)p 1 " + 3yy = o, 

which being the same equation as that deduced above, it follows that 
the points of maximum and minimum curvature are the same as those 
at which the contact is of the third order. 

(97.) In the preceding investigations we have always considered 
x to be the independent variable, because the expression for the ra- 
dius of curvature has been obtained conformably to this hypothesis. 
But if any other quantity is taken for the independent variable, the 



THE DIFFERENTIAL CALCULUS. 137 

foregoing expression for r will not apply ; therefore, in order to give 
the greatest generality possible to the formula for the radius of cur- 
vature, we shall now suppose any arbitrary quantity whatever to be 
the independent variable, x and y being functions of it. Hence, in- 
stead ofp' and p", we shall have (66) 

(dy) (d 2 y)(dx)-(d 2 x)( dij) 

(dx) {dx 3 ) 

the parentheses intending to intimate that the independent variable, 
according to which the differentials of the functions x, y are taken, is 
arbitrary, and the differential of which when chosen is with its proper 
powers to be introduced as denominators of the above differentials. 
Making, therefore, these substitutions in the expression for r, it be- 
comes 

( W + (dx) 2 )f 



(d 2 y) {dx) - (d?x) (dy) 
or, since (86) 



(ds)=V{dy) 2 +(dxy 
whatever be the independent variable, 

W 3 



(1). 



(d 2 y) (dx) - (d 2 x) (dy) 

(98.) This expression is of the utmost generality, and will furnish 
a correct formula for every hypothesis respecting the independent va- 
riable. Thus, if x be chosen for the independent variable, then 
(dx) = 1 and (d 2 x) = 0, and the formula in that case is 

ds* 

dx 2 
being the same as that at first given as it ought to be. If y be the 
independent variable, then (dy) = 1 and (d 2 y) = 0, so that upon this 
hypothesis the formula is 

ds* 

-=-!■■•■ <»>• 

dy 2 
18 



138 THE DIFFERENTIAL CALCULUS. 

If 5 be the independent variable, then (ds) = 1 ; 

... (<*V) = d SJWWyT = .'. (d 2 y) =- { ^L(d 2 x) . . . (4). 

substituting this in the denominator of (1) we have 

dy_ 

— M^ + ^=-wr-?k •••• (5) - 

da? 
By squaring (1) on this last hypothesis we have 

1 

(dhj) {dx) - (d?x) (dy) ) 2 

but, since from (4) 

(d 2 y) (dy) + (d 2 x) (dx) = 0, 
it may be added to the denominator of this expression for r 2 without 
affecting its value, so that 

1 



r> = 



(dhj) (dx) - (d 2 x) (dy) Y + (dhj) (dy) + (d 2 x) (dx) } l 
1 



(dyY + (dx) 2 X (d 2 y) 2 -\-(d 2 xf 
1 1 



(d 2 y) 2 +(d 2 x) 2 j^TTs:, 

V{ ds 2) { ds 2) 



(6). 



(99.) We shall now proceed to determine a suitable formula for 
polar curves. 

If the circle whose equation is (1) p. 131, be transformed from 
rectangular to polar coordinates, the pole being at the origin of the 
primitive axes, and the axis of x being the fixed line from which the 
variable angle w of the radius vector y is measured, we shall have 
(Anal. Geom.) 

(y cos. w — a) 2 + (y sin. w — /3 2 ) = r 2 . . . . (1). 
If, therefore, we differentiate on the supposition that w is the inde- 
pendent and y the dependent variable, and denote the first and second 
differential coefficients by p, and p in we shall have 



THE DIFFERENTIAL CALCULUS. 139 

(ycos.w — a) (pjcos.o) — ysin.t l ))-j-(ysin.w — /?) (^ sin. a) 4- y cos. w) = 0.. . (2) 
( p, cos. 03 — y sin. w) 2 -{- (y cos. <o — a) ( p t/ cos. w — 2^ sin. w — y cos. w) -f- 
(jO/sin.w-j-ycos. w) 2 +(ysin. a) — /?) (^sin. co-f^jt^cos. o> — ysin. w) =0...(3) 

If from the two latter equations we determine the values of y sin. 
w — (3 and y cos. w — a, and substitute them in (1), we shall obtain 
the following expression for r in functions of y and its differential 
coefficients, viz. 

if + p/) j_ 

r 2 + %>, 2 -yp„ ■• 

But we shall arrive at this expression more readily by first deducing 
from the equations 

y = y sin. w, x — y cos. u> 
the differential coefficients 

-j- = y cos. cj -\-p t sin. w = (c%) 
-j— = — 7 sin. w + p, cos. w = (da?) 

-7^- = — y sin. go + 2p t cos. w + p,, sin. w — (e%) 

d 2 x 

-j-g- = — 7 cos. u — 2p 7 sin. co + P// cos. co = (d 2 x) 

and then substituting them in the general formula (1). . 
Since (80) the expression for the normal PN is 

N = y 2 + ^ 2 )i 
we may put the above expression for r under the form 

N 3 
r ~ r 2 + 2p? — y Pi/ ' ' ' ' (5) - 

5. To determine the radius of curvature at any point in the loga- 
rithmic spiral 



V. 



7 


= 


60 




dy 




CO 

a 


_ 7 _ 


du 




m 


m 


d 2 y 
du 


= 


7 
m 2 


= P t r 



140 THE DIFFERENTIAL CALCULUS. 

Hence 



K/ ^ Ft > = (y 2 + V 2 ) 2 = yy/ 1 + — 

f + 2p 2 — yp K7 V,) 7V t 



l i 
y y/\ -\- . ( ar t, 80) = 7 cosec. P. 

tan*" IT 

It appears, therefore, that the radius of curvature is always equal to 
the normal. 

6. To determine the radius of curvature at any point in the curve 
whose equation is 

y = 2 cos. w ± 1 



\ r — 



(5 ± 4 cos. w) 1 
9 ± 6 cos. w 



3 



CHAPTER IXZ. 

ON INVOLUTES, EVOLUTES, AND CONSECUTIVE 
CURVES. 

(100.) If osculating circles be applied to every point in a curve, 
the locus of their centres is called the evolute of the proposed curve, 
^his latter being called the involute. 

The equation of the evolute may be determined by combining 
the equation of the proposed curve with the equations (2), (3) p. 131, 
containing the variable coordinates a, (3 of the centre. As these 
three equations must exist simultaneously for every point of contact 
(x, ?/), the two quantities x, y may be eliminated, and therefore, a 
resulting equation obtained containing only a and (3, which equation 
therefore will express the general relation between a and f3 for every 
point (x, y) ; in other words, it will represent the locus of the centres 
of the osculating circles. 

Or, representing the equation of the proposed curve by y = F#, 
we shall have to eliminate x and y from the equations (p. 131) 



THE DIFFERENTIAL CALCULUS. 



141 



y = F*, • 

when the resulting equation in a, (3 will be that of the evolute. 



EXAMPLES. 



(101.) To determine the evolute of the common parabola 



.♦. l+p' 2 = 



c .\ p = 

y 2 + 4m 2 



2m 



p" = - 



4m 3 



= 1 



m 



a = ar -f- -^— + 2m = 3# + 2m .-. x = 
2m 



. r 

2m 
a — 2m 



= s-— . 



4m 2 



_ — ?/ 3 _ — 2^ 



m 



m 2 



~ \ 4 ) 



•'•^ 2== 27 (a ~ 2w)3 ' 
which is the equation of the evolute. If the origin be removed to 
that point in the axis of x whose abscissa is 2m, then the equation be- 
comes 

The locus of which is called the semicubical parabola. 
It passes through the origin because (3 = when 



a = ; therefore the focus of the proposed involute is 



in the middle, between its vertex and the vertex of the 
evolute. (Anal. Geom. art. 100.) The curve con- 
sists of two branches symmetrically situated with respect to the axis 
of x or of a, and lies entirely to the right of the origin, for every posi- 
tive value of a gives two equal and opposite values of /3, and for 
negative values of a, (3 is impossible. It is easy to see, there- 
fore, that the form of the curve is that represented in the margin. 
2. To determine the evolute of the ellipse. 
By example 2, page 133, we have 




142 



THE DIFFERENTIAL CALCULUS, 



v- 



b 2 x , 



ahf 



, 2 _ aY -f 6 4 x 2 p f _ xf 
b* 



'. 1 +p 

a*y* 

xitfyt + Wx 2 ) 



P 



tfb* 



y(a*y 2 + b*x 2 ) 



Now, since, by the equation of the curve, 

tftf = a 2 b 2 — b 2 x 2 or tfx 2 = a 2 b 2 — a 2 xf 
.-. a'y 2 + Vx 2 = b 2 (a 4 — c 2 x 2 ) or = a 2 (6 4 + c*y 2 ), 
c 2 being put for a 2 — b 2 . Hence, by substitution, 
c 2 x 3 _ c 2 iP 



n 2 n z l 



S 2 b B i 



Substituting these values in the equation of the involute, we have 



V x 



2/y.a J. 



,a?a 



a 2 (—)* + b 2 (—y = a 2 b\ 

a 2 b 2 
or, finally, dividing all the terms by — -, we obtain for the evolute the 

C3 

equation 

(6/3)* + (aa)s = C 3 z=z (y _ 6 2 )i 

c 2 

If a = 0, then (3 = ± —, so that the curve meets 

6 

the axis of y in two points, c, d, equidistant from 

c 2 
the origin 0. If (3 = 0, then a = ± — , so that 

it also meets the axis of x in two points, b, a, equi- 

c 2 
distant from 0. If a is numerically greater than — the ordinates be- 




come imaginary, and if (3 is numerically greater than -r-the abscissa 

becomes imaginary ; therefore the curve is limited by the four points 
a, 6, c, d, and touches the axes at those points. It consists, there- 
fore, of four branches symmetrically situated as in the figure. 



THE DIFFERENTIAL CALCULUS. 143 

3. To determine the evolute of the rectangular hyperbola, its 
equation between the asymptotes being xy = a 2 . 
The equation of the evolute is 

2 
2 2 a 3 

( a +/3) 3 --( a -/3) 3 =:-. 

4* 



THEOREM. 

(102.) Normals to the curve are tangents to the evolute. 

Let the equations of the curve and of its evolute be 
y = Fx and (3 = fa, 
then differentiating the equation (2) p. 131, considering a, (3 as va- 
riables as well as x, y, we have 



but (130) 



w '2 

y-l3 = . P 



P 

Hence, by substitution, 

da t d(3 _ 

dx dx 

d(3 da d/3 1 13 — y, n , ' x 

... -±L -^ — or -f- = — — = ^ 2 (equa. 2, p. 131 . 

aa? dx da p a — x 

Now ■— — expresses the trigonometrical tangent of the angle be- 
et a 

tween the axis of x and a linear tangent through any point (a, (3) of 

the evolute, and expresses the trigonometrical tangent of the 

angle between the axis of x and a normal at any point (#, y) of the 
involute ; but this normal necessarily passes through a point (a, (3) 
of the evolute, and, therefore, in consequence of the above equality, 
it must coincide with the tangent at that point. 

THEOREM. 

(103.) The difference of any two radii of curvature is equal to the 
arc of the evolute comprehended between them. 
Differentiating the equation 



144 THE DIFFERENTIAL CALCULUS. 

on the hypothesis that a is the independent variable, we have 






but by last article 



(3 = (x — a) — - 
aa 



and 



d(3 2 
■.(*-a) 2 (£,+ l)=r*...(l), 



-<—)«£ +D=r*... (2). 



Dividing (2) by the square root of (1) we have 

da 2 da' 



that is (86) 



ds dr 

— = — .•. — s=r ± a constant, 

aa aa 



/7o //■y 

for otherwise there could not be — -r- = ^-. 

aa aa 

Hence if r, v> be the radii of curvature of any two points, and s, s' 
the corresponding arcs of the evolute, then 

r ± const. = — s 
r' db const. = — s' 



so that the difference of the two radii is equal to the arc of the evolute 
comprehended between them ; therefore, if a string fastened to one 
extremity of this arc be wrapped round it and continued in the direc- 
tion of the tangent at the other extremity as far as the involute curve, 
the portion of the string thus coinciding with the tangent will by (102) 
be the radius of curvature at that point P of the involute curve which 
it meets, and, consequently, by the above property, if the string be 
now unwound, P will trace out the involute. 



THE DIFFERENTIAL CALCULUS. 145 

On Consecutive Lines and Curves. 

(104.) Every equation between two variables may always be con- 
sidered as the analytical representation of some plane curve, given in 
species by the degree of the equation, and determinable both in form 
and position by the constants which enter it, provided, these 
constants are fixed and determinate. If, however, the equation con- 
tains an arbitrary or indeterminate constant a, then, by assuming dif- 
ferent values for a the equation will represent so many different curves 
varying in form and position, but ail belonging to the same familij of 
curves. 

Now if we consider the form and position of one of these curves to 
be fixed by the condition a = a', another, intersecting this in some 
point (a? 7 , y'), may be determined from a new condition a = a' + h ; 
and if h be continually diminished, this latter curve will approach more 
and more closely to the fixed curve, and will at length coincide with 
it. During this approach, the point of intersection (x, y') necessarily 
varies, and becomes fixed in position only when the varying curve 
becomes coincident with the fixed curve. In this position the point 
is said to be the intersection of consecutive curves, so that what mathe- 
maticians call consecutive curves, are, in reality, coincident curves, 
and the point which has been denominated their point of intersection 
may be determined as follows : 

(105.) Let 

F(x,y,x') =0 (1) 

represent any plane curve, x being a parameter, and for any inter- 
secting curve of the same family let x' become x' + h, then, since 
however numerous these intersecting curves may be, the x, y of the 
intersections belong also to the equation (1) ; it follows that as far as 
these points are concerned, the only quantity in equation (1) which 
varies is x', therefore, considering x, y as constants in reference to 
these points, we have, by Taylor's theorem, 



F(x 


t*h 


x' + h) = F ( 


r, y, x') 


-f 


dF (x, y 9 

dx' 


X>) h + 






<PF (x, y$ tf) 

dx' 2 


I • 2 


U- 


&c. 




F (a?, y, 


x') 


= 0, therefore 


19 









146 THE DIFFERENTIAL CALCULUS. 

F (a?, y, x' + h) d¥ (x, y, x') d 2 F (a?, y, x') h 



+ &c. 



h dx' dx' 2 1 -2 

hence, when the curves are consecutive, that is when h == 0, we have 
the following conditions, viz. 

F (x, y, x 1 ) = \ 

dF (x, y, x') = I .... (2) 

da?' ' 

to determine a? and y. 

Suppose, for example, it were required to determine the point of 
intersection of consecutive normals in any plane curve. 

Representing the equation of the curve by 

y' = Fx\ 
and any point in the normal by (ar, ?/), we have for the equation of the 
normal 

y — y' = j- (x — x') or (y — y') p' + x — x' — 0. 

This corresponds to the first of equations (2), a?' being the parameter ; 
hence, differentiating with respect to x of which y' is a function given 
by the equation of the curve, we have 

{y-y')p"—p' 2 -i =o 

•••*-* — ^ — 

hence (92) consecutive normals intersect at the centre of curvature, 
(106.) If we eliminate the variable parameter a/ by means of the 
equations (2), the resulting equation will belong to every point of in- 
tersection given by every curve of the family 

F (*„ y,x,x>) =0 . . . . (1), 

and its consecutive curve ; for whatever value we suppose x' to take 
in the equations (2), the result of the elimination will obviously be 
always the same. Hence this resulting equation represents the locus 
of all the intersections, and we may show that at these same inter- 
sections this locus touches every individual curve in the family. The 
equation (1), where x' represents a function of x, y, determined by 



THE DIFFERENTIAL CALCULUS. 147 

the second of the conditions (2) in last article, is obviously the equa- 
tion of the locus of which we are speaking, and the same equation, 
when #' takes all possible values from to ± oo, furnishes the family 
of curves, which we are now to show are all touched by this locus. 
Taking any one of this family, and differentiating its equation (l), x' 
being constant, we have 

, du _ du , 
du = — - ax + — du = 0. 
dx dy J 

Differentiating also the equation (1) of the locus, x' being given by 
the second condition of (2) in last article, we have 
du du , , du _ . 

du = di dx + 7hj d,J + T^ dx =0 ' 

but by the condition just referred to -— - = at the point where the 

dx 

curves whose equations we have just differentiated meet; hence, 

since at those points each of these equations give the same value for 

dy 

-p, it follows that they have contact of the first order ; we infer, 

therefore, that the equation (1), when x' is determined from the second 
of the conditions (2) last article, represents a curve which touches and 
envelopes the entire family of curves represented by equation (1), x' 
being any arbitrary constant. Thus, as we already know, the locus 
of the intersections of normals with their consecutive normals is a 
curve which touches them all at their points of intersection, being the 
evolute of the curve to which the normals belong. 

The following examples will further illustrate this theory. 

EXAMPLES. 

(1 07. ) 1 . To determine the curve which touches an infinite series 
of equal circles, whose centres are all situated on the same circum- 
ference. 

Let the equation of the fixed circle be 
a* + ^i = r \ 
then, for the coordinates of the centre of any of the variable circles, 
the expressions will be 



x and s/r' 2 — x'*, 



148 THE DIFFERENTIAL CALCULUS. 

so that the general representation of these circles will be 



( X — x 'f + (y — Vr* — x' 2 ) 2 — r 2 = = u (1), 

x' being considered as an arbitrary constant. If, however, x' be con- 
sidered not as an arbitrary constant, but as a function of x and y, 

fulfilling the condition — = 0, then, by the preceding theory, (1) 

CiX 

will represent the curve which touches all the circles in those points 
where each is intersected by its consecutive circle. Hence, differ- 
entiating (1) with respect to x, we have 

- = -(x-x) + —==-x' = 



dx sf 



'. — x Vr' 2 — x' 2 -\-x'y = 
r'x 



Vx 2 + if 
This, then, is the function of x, y, which, substituted for x, in (1), 
gives the equation of the locus sought. The result of this substitu- 
tion is 



a* + y 2 — 2r l Vx 2 -f f + r' 2 = r 2 , 
or, extracting the root of each side, 



Jo* + f = r ' ± r .'. x 2 + f == (r ± r) a , 

an equation representing two circles, whose radii are respectively 
r + r and r — r. Hence the series of circles are touched and 
enveloped by two circular arcs, having these radii, and the same centre 
as the fixed circle. 

2. Between the sides of a given angle are drawn an infinite 
number of straight lines, so that the triangles formed may all have 
the same surface, required the curve to which every one of these lines 
is a tangent. 

Let the given angle be 6, and, taking its sides for axes, we have, 
for the equation of every variable line, 

y = ax + /3 . . . . (1), 

and, putting successively y = and x = 0, the resulting expressions 
for x and y denote the sides of the variable triangle, including the 



THE DIFFERENTIAL CALCULUS. 149 

(3 
given angle, so that these sides are and (3 ; hence, calling the 

constant surface s, we have 



(3 



s = — — - sin. 6 .*. a, ; 

2a 2s 

hence the equation (1) is the same as 

/3 2 sin. 6 p 

y = ^— x + £ • • • • (2), 

where /3 is considered as an arbitrary constant. But if for this arbi- 
trary constant we substitute the function of x, arising from the condition 

— \- — 0, then (2) will represent the locus of the intersections of 
dp 

each variable line, with its consecutive line, which locus touches 

them all. Differentiating them with regard to /3, we have 

/3 sin. "','"•„ '"„ s 

— £ x+ 1 =0 .'. ft = r— , 

s # sm. 6 

this substituted in (2) gives for the equation of the sought curve 

s 



V = — : X + : r, 

° 2a? 2 sin. 6 x sm. 6 

or rather 



xy 



-2 sm.d 

hence the curve is an hyperbola, having the sides of the given angle 
for asymptotes. 

3. The centres of an infinite number of equal circles are all 
situated on the same straight line : required the line which touches 
them all 1 

JLns. They are touched by two parallels to the line of 
centres. 

4. From every point in a parabola lines are drawn, making the 
same angle with the diameter that the diameter makes with the tan- 
gent : required the line touching them all 1 

Ans. They are touched by a point, viz. the focus, in 
which therefore they all meet. 



150 THE DIFFERENTIAL CALCULUS, 



CHAPTER IV. 

ON THE SINGULAR POINTS OF CURVES, AND ON 
CURVILINEAR ASYMPTOTES. 

Multiple Points. 

(108.) If several branches of a curve meet in one point, whether 
by intersecting or touching each other, that point is called a multiple 
point. In the former case the point is said to be of the first species, 
and in the latter of the second species, and we propose here to in- 
quire how, by means of the equation of any curve, these points, if any, 
may be detected. 

Multiple points of the first species. When the curve has multiple 
points of the first species, we readily arrive at the means of determin- 
ing their position from the consideratioa that at such points there must 
be as many rectilinear tangents as there are touching branches, and, 

cly 
consequently, as many values for -^, the tangent of the inclination 

of any tangent through the point (x, y) to the axis of x ; so that the 
equation of the curve being freed from radicals and put under the 
form 

F (*, y) = 0, 
its multiple points of the first species will all be given analytically 
by the equation 

f _ du . du _ 
^ dx dy 0' 

so that no systems of values for x and y can belong to multiple points 
of the first species, but such as satisfy the conditions 

^ = 0,^ = 0, 
dx dy 

as well as the equation of the curve. Having, therefore, determined 
all such systems of values by solving the two last equations, the true 
values of j/ for each system will be ascertained by proceeding as in 
(41), and those systems only will belong to multiple points of the 



THE DIFFERENTIAL CALCULUS. 151 

first species that give multiple values to p'. Let us apply this to an 
example or two. 

EXAMPLES. 

(109.) 1. To determine whether the curve represented by the 
equation 

aif — xhj — bx 3 = 0, 
has any intersecting branches 

At the points where branches intersect we must have 
3x? (y + b) = 0, Say 2 — x 3 = 
.-. x = 0, y = 



x = %/ 3ab 2 , y — — 6 ; 
this seeond system of coordinates do not satisfy the proposed equa- 
tion, and therefore do not mark any point in the curve ; the first sys- 
tem, which is admissible, shows that if there exist any multiple point 
it must be at the origin. Hence, to ascertain the true value of p' at 
this point, we have, by differentiating both numerator and denomina- 
tor in the expression 

L^J L ^axf—x 2 J 

= 6x (y + b) + Sx 2 p' _ 
L 6aijp> — 3a: 2 -" ~~ 
_ My + b) + Up'x + Sx*p" 66 ; ,-. 3 A 

L 6ayp" + 6ap' 2 — 6x J 6a[ p'J ' ' lP J V a ' 
therefore, as this has but one real value, the curve has no intersecting 
branches. 

2. To determine whether the curve represented by the equation 
# 4 + 2axhj — ay 3 = 
has intersecting branches 

— = 4x(x 2 +ay) = 0,j- = a (2-r 2 — Sy 2 ) = 0. 



152 



THE DIFFERENTIAL CALCULUS. 



There is but one system of values that can satisfy these three equa- 
tions, viz. 

x = 0, y = 0, 
so that if there are intersecting branches they must intersect at the 
origin. To determine, therefore, whether at this point p' has multiple 
values we have 



lF J l a {'Sy 2 — 2x) 2J 



_ 6X 2 + 2ay + 2axp' 
3ayp' 

_ 4 « [>'] 



2a x 




3a[p'f — 2a 
... Za\_p'Y— 6a[p'~\ ■= 
... |y] = Oor [>'] = ± V 2 
hence //iree branches of the curve intersect at the 
origin ; the tangent to one of them at that point is 
parallel to the axis of x, and the tangents to the 
other two are symmetrically situated with respect 
to the axis of?/, since they are inclined to the axis 
of Xj at angles whose tangents are + V 2 and — V 2. 

(110.) Should the values of p corresponding to any values of a: 
and y, which satisfy the equation of the curve, be all imaginary, we 
must infer that, although such a system of values belong to a point 
of the locus, yet that point must be detached from the other points of 
the locus, for since, if the abscissa of this point be increased by h, 
the development of the ordinate will agree with Taylor's develop- 
ment, as far, at least, as the second term for all values of h, between 
some finite value and 0, it follows that all the corresponding ordi- 
nates between these limits must be imaginary, so that the proposed 
point is isolated, having no geometrical connexion with the curve, 
although its coordinates satisfy the equation. Such a point is called 
a conjugate point. 

(111.) From what has now been said, it appears that, by having 
the equation of a plane curve given, those points in it where branches 
intersect, as also those which are entirely detached from the curve, 
although belonging to its equation, may always be determined by the 



THE DIFFERENTIAL CALCULUS. 153 

application of the differential calculus, and independently of all con- 
siderations about the failing cases of Taylor's theorem, except, in- 
deed, those connected with the theory of vanishing fractions. We 
shall now seek the analytical indications of 

Multiple Points of the Second Species. 

(112.) The second species of multiple points, or those where 
branches of the curve touch each other, the differential calculus does 
not furnish the means of readily determining from the implicit equa- 
tion of the curve. We know that at such a point, p f cannot admit of 
different values, since the branches have one common tangent ; and 
we know, moreover, that if Taylor's theorem does not fail at that 
point, we shall, by successively differentiating, at length arrive at a 
coefficient which, being put under the form £, the different values will 
indicate so many different touching branches ; for if no coefficient gave 
multiple values for the proposed coordinates x\ y, then the ordinates 
corresponding to the abscissas between the limits x and x ± h, h 
being of some finite value, would each have but one value, and, there- 
fore, different branches could not proceed from the point (x\ y'). But 
we have no means of ascertaining a priori which of the coefficients 
furnishes the multiple value. When, however, the equation of the 
curve is explicit, then the multiple points of either species are very 
easily determined. Thus, if the equation of the curve be 

y — {x — a) 2 %/ x — b + c, 
we at once see that x = a destroys the radical in y and p\ that re-ap- 
pears inp" ; therefore, at the point corresponding to this 
abscissa, there will be but one tangent, and yet two 
branches of the curve proceed from it on account of the 
double value ofp". Hence the point is a double point of the second spe- 
cies, the branches have contact of the first order, and, because p' == 0, 
the common tangent is parallel to the axis of the abscissas ; if the radical 

had been of the third degree, the point corresponding to the same abscissa 
would have been a triple point, &c. It appears, therefore, that when 
the equation of the curve is solved for y, there will exist a multiple 
point, if in the expression for x a radical is multiplied by the factor 
(x — a) m . If m = 1, the branches of the curve intersect at the point 
whose abscissa is x = «, because then p at that point takes the same 

20 



154 THE DIFFERENTIAL CALCULUS. 

values as the radical, but if m > 1 then the branches touch, because 
then the radical is destroyed in^' for x = a ; in both cases the index 
of the radical will denote the number of branches which meet in the 
point. Such, therefore, are the geometrical significations of the cases 
discussed in (75) and (76). 

Cusps, or Points of Regression. 

(113.) A cusp or point of regression is that particular 

kind of double point of the second species in which the 

two touching branches terminate, and through which they 

do not pass, so that on one side of such a point, viz. on 

, that where the branches lie, the ordinate has a double 

3 value, and on the other side the contiguous ordinate has 

an imaginary value. 

The cusp represented in the first figure, where the branches are 

one on each side of the common tangent, is called a cusp of the first 

kind, and that in the second figure, where the branches are both on 

one side, a cusp of the second kind. 

(114.) It is obvious that cusps can exist only at those points, the 
particular coordinates of which cause Taylor's theorem to fail, for if 
Taylor's theorem did not fail at such a point, then the ordinates in the 
vicinity, corresponding both to x + h and to x — /i, would be both 
possible or impossible at the same time. We are not, however, to 
infer that when the adjacent ordinates are real on the one side of any 
point, and on the other side imaginary, that a cusp necessarily exists 
at that point, for it is plain that the same analytical indications are 
furnished by the point which limits any curve in the direction of the 
axis of x, or at which the tangent is perpendicular to that axis, as in 
the third figure. It becomes important, therefore, in seeking parti- 
cular points of curves to be able to distinguish the point which limits 
the curve in the direction of the axes from cusps. 

(115.) Now at the limits, the tangents to the curve are parallel to 

du 
the axes, the limits are therefore determined by the equations -j- = go 

QjX 

ana - .J. = o, and they fulfil, moreover, the following additional con- 
dx 



THE DIFFERENTIAL CALCULUS. 155 

ciitions, viz. 1°, the ordinate or abscissa, whichever it A 



> 



V 



may be, that is parallel to the tangent, immediately be- 
yond the limit, must be imaginary ; but if it be ascertained 
that this is not the case, the point is not a limit but a cusp 
of the first kind, posited as in the annexed figures, or 
else a point of inflexion ; the latter when the contiguous 
ordinates are the one greater and the other less than that at the 
point. 2°, Besides the first condition there must exist also w 
this, viz. that immediately loithin the limit the double ordinate r 
or abscissa, whichever may be parallel to the tangent, must 
have one of its values greater and the other less than at the 
point, but if both are greater or both less the point is not a limit 
but a cusp of the second kind, posited as in the annexed figures. 
Hence, when the branches forming the cusp touch the abscissa or the 
ordinate of the point, they may be discovered by seeking among the 

values which satisfy the equations — - = and — - = oo , those which 

(XX CLX 

do not fulfil both the foregoing conditions. Let us illustrate this by 
examples. 

EXAMPLES. 

(116.) 1. To determine whether the curve whose equation is 
{y-bf={x-af 
has a cusp at the point where the tangent is parallel to the axis of y. 
By differentiating 

dy _ 2 x — a 

dx~~3' (y —b ) 2 
this becomes infinite for y = b, therefore the point to be examined is 
(a, 6). In order to this, substitute a ± h for x, in the proposed equa- 
tion, and we have, for the contiguous ordinates, 

y = b± ftf, 
which is not imaginary either for + h or — h ; the point 
(a, b) -is therefore a cusp of the first kind, and posited as 
in the figure, since the contiguous values of y are both 
greater than b. 

2. To determine whether the curve whose equation is 



f 




156 THE DIFFERENTIAL CALCULUS. 

y — a = (x — 6)3 + (x — b)* 
has a cusp at the point where the tangent is parallel to the axis of y. 

Here the coefficient -~ becomes infinite for x = 6, therefore the 
ax 

point to be examined is (6, a). Substituting b -f h for x, we have 

?/ = a + p" + &*. 
For negative values of h this is imaginary, therefore the curve lies 
entirely to the right of the ordinate y = a, so that the condition 1° 
pertaining to a limit is fulfilled. To the right of this or- 
dinate the two values of?/, corresponding to a value of h 
ever so small, are both greater than y = a, so that the 
condition 2° is not fulfilled, the point (6, a) is therefore 
a cusp of the second kind, and posited as in the cut. 

3. To determine the point of the curve whose equation is 
(y-a-xT = (x-b)\ 
at which the tangent is parallel to the axis of y. 

The differential coefficient becomes infinite for x = b, therefore 
the point to be examined is (b, a + b). Substituting b + h for x, 

y = (a+b) + fc* + h 9 

negative values of h render this imaginary, therefore the 
condition 1° is fulfilled; positive values give two values 

3 

for ?/, and as h maybe taken so small that A* may exceed 

h, and since, moreover, the two values of /i 4 are the one positive and 
the other negative, it follows that the real ordinate contiguous to the 
point has one value greater, and the other less, than that at the point 
of contact ; hence the condition 2° is also fulfilled, and thus the point 
marks the limit of the curve, which, therefore, lies to the right of the 
ordinate, through x = b. 

(117.) Having thus seen how to determine those cusps where the 
branches touch an ordinate or abscissa, we shall now seek how to 
discover those at which the tangent is oblique to the axes. The true 
development of the ordinate contiguous to such a cusp must be of the 
form 

y >+%h+ A/i a +B/^-{-&c. 
dx 



THE DIFFERENTIAL CALCULUS, 157 

and the corresponding ordinate of the tangent will be 

hence, subtracting this from the former, we have 

A = A/i a + /3/iZ 3 + &c. 

(118.) Now in order that the point (#', y') may be a cusp, this dif- 
ference for a small value of h must have two values, and to be a cusp 
of the first kind these two values must obviously have opposite signs ; 

but since h may be so small that A/i may exceed the sum of all the 

following terms, h must have two opposite values ; hence, a must 
be a fraction with an even denominator, and, conversely, if a be a 
fraction with an even denominator, the point (#', y') will be a cusp of 

the first kind. Hence, at such a point, -~ is either or oo : if 

j3 >2, and oo if/3 < 2. 

(119.) In order that the cusp may be of the second kind, both 

values of A must have the same sign there, for h cannot admit of 
opposite values of the same value of h, consequently a must in this 
case be either a whole number, or else a fraction with an odd denomi- 
nator ; and conversely, if a be either a whole number, or a fraction 
with an odd denominator, the point (x, y) will be a cusp of the second 
kind, provided, of course, that A has two values. The position of the 
branches will depend on the sign of A. 
We shall now give an example or two. 

(120.) 4. To determine whether the curve whose equation is 

9 

y = x ± x 2 
has a cusp. 

Here y is possible for positive values of x, and imaginary for all 
negative values ; hence there may be a cusp at the origin. To as- 
certain this, put h for x, in the equation, and we have, for the con- 
tiguous ordinate, the value 

y = h± ifi. 



158 THE DIFFERENTIAL CALCULUS. 

The coefficient of h being 1 == — , we see that the tan- 
° dx' 

gent to the curve at the origin is inclined at 45° to the 

axes, and, since f has an even denominator the origin is a cusp of the 

first kind. 

5. To determine whether the curve whose equation is 

5 

y — a = x + bx 2 + ex 2 
has a cusp. 

Here y is imaginary for all negative values of x, therefore the point 
(0, a) may be a cusp. Substituting h for x, we have 

y = a + h + bh 2 + ch*. 
,// As before, the tangent is inclined at 45° to the axes, 

and, since the exponent of the third term is a whole num- 
ber, and the whole expression admits of two values, in 
j> 
consequence of the even root h 2 , it follows that the proposed point 

is a cusp of the second kind. The branches are situated to the right 
of the axis of y, because h must be positive, and they are above the 
tangent because bh 2 is positive. 

6. To determine whether the curve whose equation is 

{2y + x + 1) 2 = 2(1— xf 
has a cusp. 

Here values of x greater than 1 are obviously inadmissible, and 
to this value of x corresponds y = — 1 ; hence the point having these 
coordinates may be a cusp. Substituting 1 + h for x, we have 

y = — i + i * + ** 

therefore the tangent to the curve at the proposed point has the tri- 
gonometrical tangent of its inclination to the axis of x equal to \ , and 
since the fraction f has an even denominator, the point 
is a cusp of the first kind. Because h is negative, the 
branches are to the left of the ordinate to the point which 
is below the axis of x, because this ordinate is negative. 

Points of Inflexion. 
(121.) Points of inflexion have been defined at (92), and we have 



THE DIFFERENTIAL CALCULUS. 159 

there shown that a point of this kind always exists when its abscissa 

causes all the differential coefficients to vanish between the first and 

the nth, provided the nth be odd and becomes neither nor go . The 

dhi 
simplest indication therefore of a point of inflexion is [~rj] =t 0» an d 

d?y 
[ ry] neither nor go ; such indications, however, cannot be fur- 
nished by any point at which the tangent is parallel to the axis of?/, 
since in this case f~— — 1 and all the following coefficients become infi- 

nite. Neither can these indications take place at any point, for which 
Taylor's theorem fails after the third term. It becomes, therefore, 
of consequence, in examining particular points of a curve, to be able 
to detect the existence of points of inflexion by some general method, 
independently of the differential coefficients beyond the first. The 
only general method of doing this is that which we have already em- 
ployed for the discovery of cusps, and which consists simply in ex- 
amining the course of the curve in the immediate vicinity and on each 
side the point in question. Points of inflexion are somewhat similar 
to cusps, each having some of the analytical characteristics common 
to both, and to the limiting points of curves as already hinted at in 
(114). But the characteristic property of a point of inflection is, that 
the adjacent ordinates on each side are the one greater and the other 
less than the ordinate at the point. This peculiarity distinguishes a 
point of inflexion from a limit, inasmuch as at a limit 
the ordinate immediately beyond is imaginary ; and 
it distinguishes it from a cusp of the first kind, in- 
asmuch as at such a cusp the adjacent ordinates 
are either both greater or both less than at the point, 
or else, as is the case when the tangent at the point is oblique to the 
axes, one of these ordinates is imaginary, the other double. We have 
then first to ascertain at what points of the curve inflexions may ex- 
ist, or to find what points are given by the conditions 

d 2 y P 

— = _ = Oorc», 

or, which is the same thing, what points are given by the separate 
conditions. 




160 THE DIFFERENTIAL CALCULUS. 

P = 0, Q = 0, 
we are then, by examining the course of the curve in the vicinity of 
each point, to determine to which of them really belongs the charac- 
teristic of an inflexion. 

Thus the means of distinguishing points of inflexion being suffi- 
ciently clear, we shall proceed to a few examples. 

EXAMPLES. 

1. To determine whether the curve whose equati on 

y = b + (x — a) 3 
has a point of inflexion where the tangent is parallel to the axis of x* 
Here 

p' = 3 (x — a) 2 , 
and when the tangent is parallel to the axis of x, p' = 0, . •. x = a 
and y = 6, at the proposed point. In the vicinity x = a + h, 

.-. y = b + h\ 
which is greater than 6, the ordinate of the point when h is positive, 
and less when h is negative ; the point (a, b) is therefore a point of 
inflexion. 

2. To determine whether the curve whose equation is ?/ 3 = x 5 or 

y = x 3 has an inflexion at any point. 



this becomes go for x = 0, therefore a point of inflexion may exist 
at the origin. Putting h for x we have 

y = h*, 

which is greater than 0, the ordinate of the point, when h is positive, 
and less when h is negative ; hence there is an inflexion at the ori- 
gin. Also the equation of the tangent being 

y = f x J , the ordinates corresponding to x = ± h 
I ^^-■-' are both less than those given by the above equa- 
ls tion ; hence the curve lies above the tangent to the 
right of the origin, and below it to the left, as in the 
figure. 
3. To determine whether the curve whose equation is 



THE DIFFERENTIAL CALCULUS. 161 



5. 

y — x = [x — a) 2 



has a point of inflexion 




this becomes infinite for x = a, therefore a point of inflexion wiat/ 
exist at the point (a, a). In the vicinity of this point x = a + h, 

.5 

.•. ?/ = a -f /i + h' s , 

which is greater than a when h is positive, and less 
Avhen h is negative ; hence (a, a,) is a point of in- 
flexion. As the corresponding ordinates of the 
tangent y = a ± /?, one, viz. y = a -\- h, is less 
than that of the curve, and the other greater ; hence 
the curve bends, as in the figure. 

On Curvilinear Jlsijmptotes. 

(122.) Two plane curves, having infinite branches, are said to be 
asymptotes to each other, when they approach the closer to each 
other as the branches are prolonged, but meet only at an infinite dis- 
tance.* 

Hence, since the expression for the difference of the ordinatee cor- 
responding to the same abscissa in two such curves becomes less 
as the abscissa becomes greater, and finally becomes 0, when the 
abscissa becomes oo , it follows that that expression can contain 
none but negative powers of x, without the addition of any con- 
stant quantity. For, if a positive power of x entered the expres- 
sion for the difference, that expression would become not but oo , 
when x = go ; and, if there were a quantity independent of x, the dif- 
ference would be reduced to this quantity, and not to 0, for x = 0. 
Hence two curves are asymptotes to each other, when the general 
expression for the difference of the ordinates corresponding to the 
same abscissa is 

A =A'aT a + B'x'P + 'C'x~? + &c (1), 

or when the general expression for the difference of the abscissas cor- 

* Sp'rals meet their asymptotic circles only after an infinite number of revolu- 
tions; these we do not consider here, having examined them at (85). 

21 



162 THE DIFFERENTIAL CALCULUS. 

responding to the same ordinate is 

A = A'y~ a + B'y'P + Wf 7 + &c (2), 

and conversely, when the curves are asymptotes to each other ; one 
or both these forms must have place. 

If for one of the curves whose corresponding ordinates are sup- 
posed to give the difference (1) there be substituted another, which 
would reduce that difference to 

B'x' 13 + cx~y + &c. 

this new curve would be an asymptote to both, and would obviously, 
throughout its course, continually approach nearer to that which it 
has been compared to, than the one for which we have substituted it 
does. In like manner, if a third curve would further reduce the dif- 
ference (1) to 

Car" ? + &c. 
this third curve would approach the first still nearer, and all the four 
would be asymptotes to each other. It appears, therefore, that every 
curve of which the ordinate may be expanded into an expression of 
the form 

y = A^ + Ba?* + . . . . A'aT* + B'x' 1 * + &c. . . . . (3). 
admits of an infinite number of asymptotes. 

Since the general expression for the ordinate of a straight line is 
y = Ax + B, for the difference between this ordinate and that of a 
curve at the point whose abscissa is x, to have the form (1), the equa- 
tion of the curve must be 

y = Ax + B + A'x~ a + B'x~& + &c (4), 

this equation, therefore, comprehends all the curves that have a rec- 
tilinear asymptote, and among them the common hyperbola, whose 
equation is 

y = ± _5 (a? _ A 2 )* = zp —x ^ I ABar 1 + &c. 

The curves included in the equation (4) are therefore called hyper- 
bolic curves. 

The other curves comprised in the more general equation (3), not 
admitting of a rectilinear asymptote, are called parabolic curves. 



THE DIFFERENTIAL CALCULUS. 163 

The common hyperbola we see by the above equation admits of 
the two rectilinear asymptotes y = ± -r- x, and of an infinite num- 
ber of hyperbolic asymptotes. 

As an example of this method of discovering rectilinear and cur- 
vilinear asymptotes, let the equation 

my 3 — xif — mx 3 
be proposed. The development of y in a series of descending powers 
of a? is (Ex. 9, p. 52,) 

in* 
y = — m - r — &c. 

therefore the curve has one rectilinear asymptote, parallel to the axis 
of x, its equation being y — — m ; the hyperbolic asymptote next to 
this, and which lies closer to the curve, is of the fourth order, its 
equation being 

yx 3 + mx 3 + m 4 == 0. 
Again, let the equation of the proposed curve be 

b 

y= 



hx~ x 4- &c (1), 



also, since 



b 2 b 2 

*? — a2 = -r ••• x = a + i • — y~ 2 + &c ( 2 )- 

y 2 2 a a ■ 

From (1) it appears that the curve has a rectilinear asymptote, coin- 
cident with the axis of x, its equation being y = ; the hyperbola 
whose asymptotes coincide with the axes is also an asymptote, its 
equation being xy = b. From (2) it appears that the curve has 
another rectilinear asymptote, parallel to the axis of y, its equation 
being x = a ; the hyperbola next to this is of the third order. If we 
consider the radical, in the proposed equation, to admit of either a 
positive or a negative value, then there will be two rectilinear asymp- 
totes, parallel to the axis of y and equidistant from it, as also two hy- 
perbolic asymptotes, symmetrically situated between the axes. 



164 THE DIFFERENTIAL CALCULUS, 



SECTION III. 

ON THE GENERAL THEORY OF CURVE SURFACES 
AND OF CURVES OF DOUBLE CURVATURE. 



CHAPTER I. 

ON TANGENT AND NORMAL PLANES. 

PROBLEM I. 

(123.) To determine the equation of the tangent plane at any point 
on a curve surface. 

Let (x', ij, z',) represent any point on a curve surface of which the 
equation is 

z = F (x, y), 
then the tangent plane will obviously be determined, when two linear 
tangents through this point are determined. Let us then consider, 
for greater simplicity, the two linear tangents respectively parallel to 
the planes of xz, zy ; their equations are 

z — z' = a(x — x')\ m 



and 

z — z > = h{y — y) 



(2), 



But since these are tangents to the plane curves, which are the sec- 
tions through (a?', y', z',) parallel to the planes of xz, zy, therefore (77) 
dz' , dz' 

Moreover the traces of the plane through the lines (1), (2), upon the 
planes of xz y zy, being parallel to the lines themselves, a and b must 
be the same in the traces as in these lines, and since they are the 



THE DIFFERENTIAL CALCULUS. 165 

same in the plane as in its traces, it follows that the equation of this 
plane must be 

z — z<=p'(x-x') + q'iy-y') .... (3), 

in which the partial differential coefficients p', q', express the trigono- 
metrical tangents of the inclinations of the vertical traces to the axes 
of x and y respectively. 

For the angle which the horizontal trace makes with the axis of x 
we have, by putting z = 0, in (3), 

P' 

tan. inc. —, 

c l 

(124.) If the equation of the surface is given under the form 

u = F (x, y, z,) = ... . (4), 
then the expressions for the total differential coefficients derived from 
u, considered as a function, first of the single variable x, and then of 
the single variable y, are (57) 

du _ du du f 

d.e * dx dz 1 
du du du 

^^ = ^ + s* =0 ' 

from which we get the values 

du du 

, _ dx , _ dy 

du du 

dz dz 

hence, by substituting these expressions in (3), the equation for the 

tangent plane becomes 

/v du . , » du , . .. du 

PROBLEM II. 

(125.) To determine the equation of the normal line at any point 
of a curve surface. 

We have here merely to express the equation of a straight line, 
perpendicular to the plane (3), and passing through the point of con- 
tact (*', y\ z'.) 



166 THE DIFFERENTIAL CALCULUS. 

New the projections of this line must be perpendicular to the traces 
of the tangent plane, or to the lines (1), (2,) hence the equations of 
these projections must be 

x — x'+ p' ( z — z') = 0) 

y — y' + q' (z — *) = oj 

which together, therefore, represent the normal. 

(126.) If we represent by a, (3, y, the inclinations of this line to 
the axes of x, y, s, respectively, then {Anal. Geom.) 

— p' 



cos. /3 = 



Vp' 2 + q' 2 + 1 

Vp' 2 +q' 2 +l 
1 



C0S - y = Vp' 2 + q' 2 +1 
(127.) If the equation of the surface be given under the form (4), 
last problem, then, in these expressions for the inclinations, we must, 
instead of p' and q', write their values as before determined from that 
equation. If, for brevity, we put 

1 



)du 2 du 2 du 2 



the expressions for the cosines will then be 

du _ du du 

cos. a = v — , cos. p = v — , cos. y = v — . 

ax dy dz 

As every plane which contains the normal line must be perpendicular 
to the tangent plane, it is obvious that there exists an infinite number 
of normal planes to any point of a surface. 



PROBLEM III. 

(128.) To determine the equation of the tangent line to any point 
of a curve of double curvature. 

We have already indicated (Anal. Geom.) how this equation is to 
be determined : 

Let 



THE DIFFERENTIAL CALCULUS. 167 

y =f X ,Z = Fx . . . . (1) 

be the equations of the projections of the proposed curve, on the 
planes of xy, xz, and let (x', y', z\) be the point to which the linear 
tangent is to be drawn, which point will be projected into (x, y') and 
(#', z',) on the plane curves (1), therefore tangents through them to 
these plane curves will be represented by the two equations 

y — y' = v' — x ') \ (2 ) 

these, therefore, together represent the required tangent in space. 

PROBLEM IV. 

(129.) To determine the equation of the normal plane at any point 
in a curve of double curvature. 

The equation of any plane passing through a proposed point is 
(Anal. Geom.) 

A(x — x') + B{y — y') + C(z — z') = 0,. . . (1), 
and for the traces of this plane on the planes of xy, xz, we have, by 
putting in succession z = 0, y — 0, the equations 

A C 

y — y = — g- 0* — *') + g- z' 

A , , , B , 

z — z' = ——(x — x') + — y', 

but since these two traces are respectively perpendicular to those 
marked (2), last problem, 

hence the equation (1) becomes 

x — x'+ 1 j'(y — y') + q ' (z~z') =0 . . . . (2), 
which represents the normal plane sought. 



168 



THE DIFFERENTIAL CALCULUS. 



CHAPTER XX. 

ON CYLINDRICAL SURFACES, CONICAL SURFACES, 
AND SURFACES OF REVOLUTION. 

(130.) These surfaces have been considered in the An aly Heal 
Geometry, and the general equations of the two first classes have 
been deduced, on the hypothesis that the directrix is always a plane 
curve. TVe shall now suppose the directrix to be any curve situated 
in space, and investigate the differential equations of these surfaces, 
as also of surfaces of revolution in general. 

Conical and Cylindrical Surfaces, 

PROBLEM I. 

To determine the equation of cylindrical surfaces in general. 
Let the equations of the generating straight line be 

x = az + a ) ( a — x — az ,.. 

y = bz+ (3$ •'' \(3 = y-bz' * * ' (1) ' 

and the equations of any curve in space considered as the directrix, 
V(x,y,z) =--0,f(x,y,z) =0 . . . . (2). 
Now for every point in this directrix, all these equations exist 
simultaneously ; moreover, the constants a, b, are fixed, since the 
inclination of the generating line does not vary, but the constants 
a, /3, are not fixed, since the position of the generating lines does 
vary. If, then, we eliminate x, y, z, from the above equations, there 
will enter, in the resulting equation, only the constants a, 6, and the 
indeterminates a, /3, hence, solving this equation, lor (3 we shall get 
a result of the form /3 = pa ; consequently, if we now substitute in 
this the values of a and (3 given above, in terms of x, y, z, we shall 
have this general relation among these variables, viz. 

y — bz = 9 (x — az) = . . . . (3), 
which is the equation of cylindrical surfaces in general, the function 
cp depending entirely on the nature of the directrix. 



THE DIFFERENTIAL CALCULUS. 169 

(131.) Now, by differentiation, this function may be eliminated 
(58), hence, 

— bp' _ * — °p' 

1 — bq' — aq ' 

, , i , -. dz dz ... 

.♦. ap + 6g = 1 or a -r- + o -y- = 1 . . . . (4), 

which is the general differential equation of cylindrical surfaces. 

(132.) The same equation may be immediately deduced from the 
general equation of a tangent plane, to the cylindrical surface. Thus, 
the equation of any tangent plane, through a point (a?', y', z) being 

z _ z ' = p ' (a; _ x ') + q > (y — y'), 

the condition necessary for it to be always tangent to the cylinder on 
which this point is situated, is merely that it may be always parallel 
to its generatrix (1), and this condition, expressed analytically, is 
(Anal. Geom.) 

ap' + bq' — 1 = ... . (4), 
this is, therefore, the relation which must have place between the par- 
tial differential coefficients derived from the equation of the surface, 
in order that that surface may be cylindrical, and it agrees with the 
relation before established. 

If, in this equation, we write for p' ', q', their values deduced from 
the equation u = of any cylindrical surface, as exhibited in (124), 
it becomes 

a c }l 4. 1 c l!i + ^ = (5) 

dx dy dz 

PROBLEM II. 

(133.) Given the equation of the generatrix, to determine the cy- 
lindrical surface which envelopes a given curve surface. 

Since the cylinder envelopes the given surface, the curve of con- 
tact is common to both, therefore every tangent plane to the cylinder 
touches the enveloped surface in that curve. The equation of any of 
these tangent planes is 

z — z'=p'{x — x) + q' (y — y), 
whether p' and q' be derived from the equation of the surface, and 
take those particular values which restrict them to the curve of con- 

22 



170 



THE DIFFERENTIAL CALCULUS. 



tact, or whether p' and q' be derived from the equation of the cylin- 
der, and preserve their general values, because in the one case the 
contact of each tangent is confined to a point in the curve of contact, 
and in the other case the contact extends along the whole length of 
the cylinder. Hence, for the curve of contact, the condition (5) must 
have place, as well as for the entire surface of the cylinder. The 
mode of solution is, therefore, obvious ; we must deduce p' and q 
from the equation of the given surface, and substitute them in (5), 
the result combined with the equation of the given surface, will ob- 
viously represent the curve for which p' and q' are common to both 
surfaces ; that is to say, we shall thus have the equations of the di- 
rectrix, and that of the generatrix being also given, the particular cy- 
lindrical surface becomes determined. 

(134.) If the proposed curve surface be of the second order, then 
the equation (5) will necessarily be of the first degree in x, y, z, and 
will, therefore, represent a plane ; so that the combination of this, 
with the equation of any surface, must necessarily represent a plane 
section of that surface ; we infer, therefore, that if any cylindrical 
surface circumscribe a surface of the second order, the curve of con- 
tact will always be a plane curve, and consequently of the second 
order, and therefore the cylinder itself must be of the second order. 

PROBLEM III. 

(135.) To determine the general equation of conical surfaces. 

Let (x\ y\ z\) be the vertex of the conical surface, then since the 
generatrix always passes through this point, its equations, in any po- 
sition, will be 

y — y'=,b(z — Z')) ' ' ' ' ^' 

x — x' y — y 

z — z z — z 

Also let the equations of the directrix be 

F(x,y,z) = 0,f(x,y,z) =0 . . . . (2), 
then, since for every point in this line, the equations (1) and (2) ex- 
ist together, we may eliminate the variables x, y> z ; the result will be 
an equation, containing the fixed constants x\ y', z' and the indetermi- 
nate s a, b ; therefore, solving this equation for b, we shall have 6 = 



THE DIFFERENTIAL CALCULUS. 



171 



(pa. Hence, substituting for a, 6, their values in terms of x, y, we 
have 

y — y x — x' 

for the equation of conical surfaces in general, the function <p depend- 
ing entirely on the directrix. 

(136.) Eliminating the function 9, by differentiating each member 
of this equation with respect to x and y, and dividing the results as in 
(58), we have 

(y — y') p' — z — z ' — ( x — x '} p' 

z — z' — (y — -y')q' (x — x 1 ) q' 

which reduces to 

z — z' = p' (x — x 1 ) + q' (y — y'), 
the differential equation of conical surfaces in general. 

(137.) This same equation, like that of cylindrical surfaces, may 
be obtained more readily by the consideration of the tangent plane, 
which, as it always passes through the vertex (x\ y\ z 1 ,) is, in every 
position, represented by the equation 

z — > z' = p' (x — x') + q' (y — y'), 
this relation, therefore, must exist between the partial differential co- 
efficients p', q', for every point of the surface, in order that it may be 
conical. 

As in Problem I. if for p\ q', we substitute their values derived 
from the implicit equation of any conical surface, the differential equa- 
tion becomes 

/ /n du ..du.. . du 

(X — X') — + (y — y')—-\-( Z — z')- r = 0. 

ax ay dz 

PROBLEM IV. 

(138.) Given the position of the vertex, to determine the equation 
of the conical surface that envelopes a given curve surface. 

Since the cone envelopes the proposed surface, the curve of con- 
tact is common to both, so that the tangent planes to the cone touch 
also the given surface, according to this curve. The equation, there- 
fore, of the tangent plane 

z — z' = p' (x — x') + q (y — y') • . • . (1), 



1*72 THE DIFFERENTIAL CALCULUS. 

holds equally for any point on the conical surface, and for any point 
in the curve of contact. Hence, if the values of p', q\ be derived 
from the equation of the given surface, and substituted in (l), this, 
combined with the equation of the given surface, must represent the 
curve common to both surfaces, that is, the directrix of the cone. 
Therefore, the vertex and directrix being known, the equation of the 
required conical surface becomes determinable. 

(139.) If the given curve surface be of the second order, the equa- 
tion (1) will be also of the second order ; but, nevertheless, the com- 
bination of these two equations will be that of a plane, for a surface 
of the second order may be generally represented by the equation 

A* 2 + By 2 + Cz 2 \ 
+ 2Dyz + 2Exz + 2Fxy } = K . . . . (2), 
+ 2Gy + 2Uy + 2Jz ) 

which gives 

dz_ _ Ax + Fy + Ez + G 

dx Ex + By + Cz + J 

dz__ Fx + By + Bz + H 

dy Ex + By + Cz + J ' 

substituting these values for p' and q', in the equation (1), and sub- 
tracting from the result the equation (2), we have, 

(Ax' + F?/ + EV + G) x -\ 
+ (Fx' + By' + Bz' + ll)y ( 

+ (Ex' + By' + Cz' + J) z ( - ° C3J, 

+ Gx' + Hy' + Jz' + K ) 

which is the equation of a plane ; therefore, the conical surface which 
circumscribes a surface of the second order, must itself be also of the 
second order. 

(140.) The above proof is from Monge (Application de V Analyse 
a la Geomeirie), but it may be rendered much more concise, by assu- 
ming the axes of reference so as to give the general equation of the 
surface a simpler form. Thus, let the axes of x pass through the 
centre, if the surface have a centre, or be parallel to its diameters if 
it have not, and let the other two axes be parallel to the conjugates 
to this, the form of the equation will then be 

Az 2 + By 2 + Cx 2 + 2F* = G (4), 



THE DIFFERENTIAL CALCULUS. 173 

dz _ F + Cx dz = By 

dx Az ' dy Az' 

These values, substituted forp' and q' in (1), convert that equation 

into 

(F + Cx)(x- x') By(y-y') _ 
z — z + - -r 7 u, 

Az Az 

or 

Az 2 -f B?/ 2 -f Cz 2 + F* — As's — By'y — Cx'x — Fx' = 0. 
The difference between this, and (4), is 

Az'z -f By'y + Cx'x + Fx + Fa?' = G, 
the equation of a plane. 

(141.) Referring again to Mongers process, we may remark, that 
if we accent the constants in the general equation (2), it maybe taken 
as the representative of another surface of the second order, for which 
the plane of contact with a circumscribing cone, whose summit coin- 
cides with that of the former cone, will be represented by the equa- 
tion 

(Ax' + Fy' + Fz' + G') x\ 
+ (Frf + By' + D** + H') j7 
+ (Ear' + By' + Qz' + J') z( ~ U * * * * ^* 

+ Gx' + W + Jz' + K') ) 

Now, although equation (3) be multiplied by an indeterminate con- 
stant, p, the result will still represent the same plane, and this plane 
will obviously be identical to that represented by (5), provided the 
coefficients of the variables x, y, z, are the same in both equations, 
that is to say, provided we have the conditions 

p {Ax' + Fy' + Fz' + G) = Ax' + Fy' + Fz' + G' 
p (Fx' + By' + Dz + H) = Fx' + By' + Dz' + H' 
p {Fx' + Dy' + Qz' + J) = Fx' + By + Gs' + J' 
p (Gx' + Fly' + Jz' + K) = Gx' + Ry' + Js' + K'. 

As, therefore, the four quantities x', y', z', p, are arbitrary they 
may be determined so that these conditions shall be fulfilled, the four 
equations being just sufficient to fix the values of these four quanti- 
ties, and as each of them enters only in the first degree, they will 
each have but one value. It follows, therefore, that there is a certain 
point, and only one, from which, as a vertex, if tangent cones be 



174 THE DIFFERENTIAL CALCULUS. 

drawn to two given surfaces of the second order, their planes of con- 
tact shall coincide The common vertex will be at the intersection 
of those diameters to each of which the plane of contact is conjugate ; 
since it has been shown above, that the vertex of the tangent cone is 
always situated on that diameter of the surface, to which the plane of 
contact is conjugate* 

(142.) We may here observe, that as we have not fixed the origin 
of the axes to any particular point on the diameter which has been 
taken for the axis of x, nor, indeed, the diameter itself, we may con- 
sider the diameter to be that passing through the vertex (x f , y', z\) of 
the cone, and this point to be the origin, in which case x\ y\ 2', will 
each be 0, and the equation of the plane through the curve of contact, 
will then be simply 

hence, the plane through the curve of contact, is conjugate to the 
diameter through the vertex of the cone. If this vertex be supposed 
infinitely distant, the same result will belong to the circumscribing 
cylinder, viz. that the plane of the curve of contact, is conjugate to 
the diameter parallel to the generatrix of the cylinder. 

Surfaces of Revolution. 

(143.) The surfaces of revolution, considered in the Analytical 
Geometry, comprise those only in which the revolving curve is always 
situated in the plane of the fixed axis. We shall here treat of sur- 
faces of revolution in general, the revolving curve being anyhow 
situated with respect to the axes. Sections of the surface, in the 
plane of the axis, are called meridians. 

problem v. 

(144.) To determine the equation of surfaces of revolution in 
general. 

Let the equations of the generating curve be 

F{x;'y,z) = 0,/ (*, y, z) = .... (1), 

* For these, and other kindred properties, the student is referred to Mr. Davies's 
paper on Geometry of Three Dimensions, in Leybourri's Repository, vol. 5. 



THE DIFFERENTIAL CALCULUS. 175 

and those of the fixed axis 

x = az + a \ /0 v 

y = bz + (3 f • ' ' " ^' 

then, since the characteristic property of surfaces of revolution is, 
that every section perpendicular to the fixed axis is a circle, we shall 
have first to determine a plane perpendicular to the line (2), and then 
to express the condition that this plane, combined with the surface, 
always represents a circle whose centre is on (2). Now the equa- 
tion of the required plane is (Anal. Geom.) 

z + ax + by = c . . . . (3), 
and the condition is, that it must give the same section as if it were to 
cut a sphere, whose centre we may fix at pleasure, but whose radius 
will vary with the section, that is, it will depend upon c in equa. (3). 
Assuming the centre of this sphere at the point where the line (2) 
pierces the plane of xy, its equation will be (Anal. Geom.) 
(*_ a y + ( ?J _ /3) 2 + z 2 = r 2 . . . . (4). 
Hence, supposing r to be the proper function of c, the equations (1), 
(3), (4), must all have place together; hence we may eliminate x, 
y, z, and thus determine what the relation between r and c must ne- 
cessarily be, to render these equations coexistent. The result of the 
elimination will obviously lead to c = cpr 2 , hence, substituting for c 
and r their values in terms of the variables, we have, finally, 

z + ax + by = 9 \(x — af + fo— f3f + z>\ .... (5), 
for the relation which must always exist among the coordinates of every 
point, in every circular section. This, therefore, is the equation of 
surfaces of revolution in general. 

(145.) If the fixed axis be taken for the axis of z, then «, a ; 6, (3, 
are each 0, therefore, in this case, the general equation becomes 

z = 9 (x 2 + f + a 2 ) • • • • (6), 
which, solved for z, takes the form 

* = + (*» + *■) .... (7). 
(146.) There is one case of this general problem, viz. that where 
the generatrix is a straight line, revolving round the axis of z, but not 
in the same plane with it, that deserves particular notice. 

Let us take, for axis of x, the shortest distance between the axis of 



176 THE DIFFERENTIAL CALCULUS. 

z and the generating line ; then this axis will be perpendicular to both 
(65), the equations, therefore, of the line will be 

x = ± a, y = ± bz, 
also, for any variable section perpendicular to the axis of z 
z = c, x 2 + ?/ 3 + z 2 = r 3 . 
Eliminating x, y, z, we have 

a 2 + b 2 c 2 + c 2 = r 2 , 
for c and r 2 putting their values above, we have 

By putting successively ^ = 0, x — in this equation, the result- 
ing forms belong to hyperbolas, hence the surface is the hyperboloid 
of revolution of a single sheet. The equation of the hyperbola cor- 
responding to x = is 

y a 2 

so that y = ± bz is the equation of the asymptotes, (see Anal. Geom.) 
hence the generating straight line, in its first position, is in a plane 
with and parallel to one or other of the asymptotes of that hyperbola 
in its first position, which would generate by revolving round the axis 
of z, the same surface as the line ; these two lines, therefore, continue 
parallel during the revolution of both; the one, viz. the asymptote, 
generating the conical surface asymptotic to the hyperboloid genera- 
ted by the other line, viz. the line 

x = ± a, y = bz, 
or 
x = ± a, y = — bz, 
and it therefore follows that these four lines will be the sections made 
on the surface by two tangent planes to the asymptotic cone drawn 
through any diametrically opposite points in its surface ; these will 
cut each other on the surface two and two, and include an angle 
equal to that between the asymptotes, so that the surface may be 
generated by the revolution of either of these intersecting lines. 

We shall shortly see that hyperboloids of one sheet, in general, 
admit of two distinct modes of generation by the motion of a straight 
line. 

(147.) Eliminating the indeterminate function 9, which depends on 



THE DIFFERENTIAL CALCULUS. 177 

the nature of the generating curve (1) by differentiation, as, in the 
preceding problems, we find 

p' + « x — a + p'z 

q'+b~y — (3 + q'z' 
from which results the partial differential equation 
(y — fi—b z )p'—( x —a.—az)q'— b(x— a) — a(y — /3) = . . (1), 

and when the axis of z coincides with that of revolution, this becomes 
y p> _ X q' = 0. 

The differential equation of surfaces of revolution may also be ob- 
tained from the consideration of the normal, which must always cut 
the axis of revolution, being situated in the meridian plane. Thus 
the equations to the normal are (125) 

x — x' + p' (z — z') = \ 

y-y' + q '(z-z>) = o$ 

and as these must exist simultaneously with the equations (2), we 
may eliminate x, y, z, and the result will necessarily be the required 
relation between p', q, and the variable coordinates x', y\ z f , of any 
point on the surface. 

PROBLEM VI. 

(148.) A given curve surface revolves round a given axis, to de- 
termine the surface which touches and envelopes the moveable surface 
in every position. 

The enveloping surface touches the moveable one in every posi- 
tion ; if, therefore, we take any particular position of the latter, their 
combination will give the curve of contact ; this curve being common 
to both surfaces, the tangent planes, at all its points, are common to 
both surfaces; hence, the values of p\ q\ which vary only with the 
tangent plane, are the same for both surfaces, as far as this common 
curve is concerned, and it is evidently by the revolution of this curve 
round the fixed axis, that the enveloping surface is generated. Hence, 
to determine this curve, we must deduce p, q', from the given equa- 
tion, substitute them in the general equation ( 1 ) of surfaces of revolu- 
tion, since there is a line on some such surface to which they belong, 
as well as to the given surface ; and then, to determine what this 
line really is, it will be necessary merely to combine this last result 

23 



178 THE DIFFERENTIAL CALCULUS. 

with the equation of the given surface : we shall thus obtain the 
equations of the generating curve, and the position of the fixed axis 
being previously known, the enveloping surface is determinable by 
Prob. Y. 

(149.) As an illustration of this, let us suppose a spheroid to re- 
volve about any diameter, to find the equation of the surface envelop- 
ing it in every position. 

Let the surface be referred to the principal diameters of the sphe- 
roid, then the equations of any other diameter will be 

x — az, y = bz . . . . (1), 
and the spheroid itself may be represented by the equation 

x 2 + y 2 + n 2 z 2 = m 2 , 
from which we derive 

, _ 1 x f _ 1 y 

P if? 9 jT'? 

substituting these values in the general equation, for all surfaces of 
revolution round the proposed axis (1), that is in the equation 

(y — bz) p' — (x — az) q' + ay — bx = 0, 
and we have 

(ay — bx) (1 — — ) = 0, 

it 

.'. ay = bx, 
hence, combining this with the given equation, we have, for the gene- 
rating curve of the envelope, the equations 

x 2 + y 2 + nV = m 2 ) ,_ 

ay = bx ) ' * * ' * * 9 

hence, the envelope itself is to be determined thus. We must elimi- 
nate x, y, z, by means of (2), and the equations 

z + ax + by = c \ ,„. 

a> + f + *■ = r * ) • • • ■• W« 

of any circular section, the result will be 

(rV __ m 2) (a 2 + b 2 ) = (c Vn 2 —l — Vm 2 — r 2 )\ 
putting for r and c their values in terms of x, y, z, we have finally, 
> 2 {x 2 + f + z 2 ) — m 2 \ (a + b 2 ), 
= \{z + ax + by)Vn 2 —l — Jm 2 — x 2 —tf — z*\\ 
which is the equation of the enveloping surface.* 

* This solution is from Hymens Geometry of Three Dimensions, p. 145. 



F/HE DIFFERENTIAL CALCULUS. 179 



CHAPTER III. 

ON THE CURVATURE OF SURFACES IN GENERAL. 

(150.) The simplest method of contemplating surfaces, is by con- 
sidering them as produced by the motion of a line straight or curved, 
which, in all its positions, is subject to a fixed law. Viewed under 
this aspect, surfaces seem to divide themselves into two distinct and 
very comprehensive classes, viz. those whose generatrices must ne- 
cessarily be curves, and those whose generatrices may be a straight 
line. If, in this latter class of surfaces, the law which regulates the 
generating straight line be such that through any two of its positions, 
however close, a plane may always be drawn, then it is obvious, that 
in every such surface, if a plane through the generatrix in any posi- 
tion, but not through any other points of the surface, that is if a tan- 
gent plane, be drawn, this plane, if supposed perfectly flexible, might 
be wrapped round the surface, without being twisted or torn, or, on 
the contrary, the surface itself might be unrolled, and would then co- 
incide in all its points with the plane. Surfaces of this kind are, there- 
fore, very properly distinguished by the name Developable Surfaces ; 
the simplest of these are the cone and cylinder. 

(151.) We see, therefore, that these surfaces are such that a plane 
may be drawn through any two positions of the generatrix, and which 
if turned round one position supposed fixed, will pass through all the 
intermediate positions of the other. But if the law of generation is 
such that this cannot have place for any two positions, however close, 
then the tangent plane, through one position, could plainly never be 
brought to pass also through another position, however near, without 
being twisted. Such surfaces, therefore, are properly designated by 
the name Twisted Surfaces. 

These two kinds of surfaces will be separately discussed hereafter, 
the particulars in the present chapter relate to curve surfaces in gene- 
ral. 

Osculation of Curve Surfaces. 

(152. Let the equations of two curve surfaces be 



180 THE DIFFERENTIAL CALCULUS. 

z =f( x i l j)i Z = F (x,y), 
when referred to the same axes of coordinates. The first of these 
surfaces we shall suppose fixed, both in magnitude and position by the 
constants a, 6, c, &c, which enter its equation, being fixed. The 
second surface we shall suppose fixed only in form, by the form of its 
equation being given, but indeterminate as to magnitude and position, 
on account of the arbitrary constants, A, B, C, &c, which enter its 
equation. 

Let now the variables x, and y, take the increments h and k, then, 
for the first surface, we have (60) 

. dz dz d 2 z d 2 z 

2' = 2 + - r fe + j-^r + i jj h 2 -f 2 — — - hk -f 
dx dy eur dxdy 

— h?) + &c. 
df } T 

and for the second, 

„ , dZ , , dZ . , ,d 2 Z , n d 2 Z .. , 
dx dy dxr dxdy 

or, more briefly, 

z' = z + p'h + q'k + \ (r'h 2 + 2s7iA; + t'W) + &c. 
Z = Z + F/i + Q'& + i (R7i 2 + 2S7iA; + Tk 2 ) + &c. 

Now the constants A, B, C, &c. being arbitrary, we may determine 
one of them in functions of x, y and the known constants, so that the 
condition 

z= Zorf(x,y) = F (x,y) 
may be fulfilled. Such a value substituted for the constant in the 
equation Z = F (x, y,) will cause all the surfaces represented by 
this equation to have a point (x, y, z,) in common with the given sur- 
face. If two more of the arbitrary constants be determined from the 
conditions 

P' = F, q = Q', 

the resulting values of these constants being also substituted in the 
same equation, the surfaces then represented will, in consequence, 
all have a common tangent plane at the point (x, y, z,) with the fixed 



THE DIFFERENTIAL CALCULUS. 181 

surface. Therefore, that this may be the case, three arbitrary con- 
stants, at least, must enter the proposed equation, and the contact 
which they determine is called contact of the first order. Contact of 
the second order requires that the following additional conditions be 
fulfilled, viz. 

r = R', s' = S', V = T", 

requiring three more arbitrary constants to be determined, and so on ; 
and that surface, all whose arbitrary constants are determined agreea- 
bly to these conditions, will, for reasons similar to those assigned at 
(87) for plane curves, touch the proposed surface more intimately 
than any other surface of the same order. It is called the osculating 
surface of that order. 

If the touching surface be a sphere, then, since in its equation there 
can enter only four disposable constants, the contact cannot be so high 
as the second order, seeing that for this there must be six disposable 
constants, but as contact of the first order would leave still one con- 
stant arbitrary, it follows that an infinite number of spheres may have 
simple contact with a surface at any proposed point, yet one of these 
may be determined that shall be strictly the osculating sphere, or 
which shall touch more intimately all round the point of contact than 
any other. 

Curvature of different Sections. 

PROBLEM I. 

(153.) At any point on a curve surface to find the radius of cur- 
vature of a normal section. 

For greater simplicity, let us suppose the plane of xy to coincide 
with the tangent plane at the proposed point, then the axis of z will 
coincide with the normal, and all the normal sections will be vertical. 
Let the plane of the proposed section be inclined at an angle & to the 
plane of xz, then the angle which its trace x' on the plane of xy makes 
with the axis of x will obviously be 6, and the x, y of this trace will 
also be the x, y of the section. Now (97) the radius of curvature p 

at the proposed point where, (86), (dx f ) = (ds) and — -j- = o, is 

\UjX J 



182 THE DIFFERENTIAL CALCULUS, 

Ok 8 df_ 

dx 2 dx 2 



But (86) 



-j-j r + 2s'-/ + t f -f- 
daf dx dx 2 



dx 2 dx 2 dx 2 



hence, by substitution, 

1 -}- tan. 2 



v r' + 2s tan. + t' tan. 2 r' cos. 2 + 2s cos. sin. + J' sin. 2 " ( '" 
For the radius of curvature p', of a second normal section inclined at 
an angle 6 + 90° to the plane of xz, we have, by putting 6 + 90° 
ford, 

. . (2), 



r r' sin. 


2 6- 


— 2s' cos 


d 


sin. a + *' 


cos. 2 0" 


.equently, 














1 

P 


+ 7 = 


V 


+ t . . . 


• (3), 



so that the sum of the curvatures of any two normal sections through 
the same point at right angles to each other, is a constant quantity. 

(154.) Consequently, when one of these curvatures is the greatest 
possible, the other must be the least possible ; that is, at every point 
on a curve surface, the sections of greatest and least curvature are al- 
ways perpendicular to each other, which beautiful theorem was first 
discovered by Euler, and is demonstrated by most writers on curve 
surfaces, though in a manner far less simple than that above. 

(155.) To determine the values of the radii of curvature of any 
perpendicular sections at their point of intersection, let the plane of 
xz be made to coincide with one of them by turning round the nor- 
mal ; that is to say, let 6 = 0, then the foregoing expressions for p 
and p' become 

(156.) But to determine the expressions for the radii of greatest 
and least curvatures, without causing the vertical planes of coordinates 
to coincide with the sections, we must know the inclinations of these 



THE DIFFERENTIAL CALCULUS. 183 

sections to the vertical planes, that is, we must know the angle 0. To 
find this from the property — = max. or min. we have, taking & for 
the independent variable in the expression for p, 

*! 

-£■ = — 2r' cos. 6 sin. & + 2s' (cos. 2 6 — sin. 2 d) + 
ad 

2? sin. cos. = . . . . (5), 

or, dividing by 2 sin. 2 0, we have 

cot. 2 6 +^^ cot. 0—1=0 (6), 

s 

from which we get for the two inclinations sought 



. r' — t'±V r' — t'f + 4s' 2 cos. 6 

cot. & = ~ — = - — r » 

2s' sm. 6 

the upper sign corresponding to the maximum, and the under to the 

minimum. Substituting these values in the first of the expressions 

(1), which may be written thus 

cot. 2 d + 1 

p ~ r' cot. 2 + 2s' cot. 6 + f 

we have for the radii of greatest and least curvatures the expressions 

2 



*"' + *'+ VV — 2 + 4*' 2 / , 7 v 

2 > • • • \ l h 

R= C 

r'+t 1 — V(r' — a + 4» y O 

These are called the principal radii of curvature at the proposed 
point, and the sections themselves the principal sections through that 
point. 

(157.) If we know the principal radii and the inclination 9 of any 
normal section to a principal section through the point, the radius of 
curvature of the normal section at that point may be expressed in 
terms of these known quantities. For, bringing the vertical coordi- 
nate planes into coincidence with the planes of principal section, we 
have = 0, and, consequently, as appears from equation (6), last 
article, s' = ; and, since (4) 

! =»4 = <. 



184 THE DIFFERENTIAL CALCULUS, 

we have 



Rr 

(8) 



r' cos. 2 9 + V sin. 2 9 r sin. 2 <p + R cos. 2 9 



p R sin. 2 9 ' r cos. 2 9 

It is plain from this expression that if R and r have the same sign, 
p will have that sign for every section through the proposed point, 
which is the same as saying that if the principal sections are both 
convex or both concave, every other section through the same point 
will be similarly convex or concave, and, therefore, also the entire 
surface at that point. In such a case the minimum radius must be 
absolutely shorter than any other radius of curvature at the point, and 
the maximum radius longer than any other. 

(158.) If the two principal radii have not only the same sign but 
the same length, then the foregoing expression gives always p = R 
whatever be the inclination 9, so that then all the normal sections 
have the same curvature and all are ■principal sections, as is the case 
with the sphere and with the ellipsoid of revolution, the paraboloid of 
revolution, &c. at those points through which the fixed axis passes. 

(159.) If the surface belong to the second of the classes mention- 
ed in (147), then no point can be assumed on it through which a 
straight line may not be drawn, and, as the curvature of this line is 0, 
it follows that the curvature of the section perpendicular to it must 
be equal to the sum of the curvatures of any two perpendicular sec- 
tions through the same point. 

(160.) Let us now suppose that the principal radii R, r have dif- 
ferent signs, as r positive and R negative, which will be the case if 
one of these sections be convex and the other concave, we shall then 
have 

_ Rr 

R cos. 2 9 — r sin. 2 9' 
which becomes infinite when 

R 

r sin. 2 9 = R cos. 2 9 or when tan. 9 = ± </ — , 

r 

but for all positive and negative values of 9 between this and 0, p will 
be positive, while beyond these limits p will be negative. 

It appears, therefore, that if from the origin two straight lines be 



THE DIFFERENTIAL CALCULUS. 185 

drawn in the tangent plane inclined to the axis of x at the angles <p = 

T> T» 

-f- V — and <p — — \f — , these will coincide with the surface ; all 
r T r 

the sections between the sides of the two opposite angles thus form- 
ed will be convex, all the sections between the sides of the other two 
opposite supplementary angles will be concave, so that the two straight 
lines which we have seen may be drawn from the proposed point to 
coincide with the surface, separate the convexity from the concavity 
at that point. 

(161.) In order to determine whether the principal radii at any point 
are both of the same sign or not, we may observe that the expressions 
(7) for these radii at art. (156) may be put under the form 

2 



r +t' + V{r' + t'f — 4 [r't — s' a ) ) 
R = 2 _ K . . . . (10), 

from which forms we immediately see that the radii will have the same 
sign, viz. positive if r't' — s' 2 > 0, and contrary signs if r't' — s' 2 < 
; this last condition, therefore, exists in the case just considered. 

(162.) We shall terminate these remarks by showing that a para- 
boloid of the second order may always be found, such that its vertex 
being applied to any point in any curve surface, the normal sections 
through that point shall have the same curvature for both surfaces. 

For, take the planes of the principal sections for those of xz, yz, 
then the radii of these sections being R, r we know that a paraboloid, 
whose vertex is at the origin, will in reference to the same axes be 
represented by the equation {Anal. Geom.) 

Z 2r ± 2R' 
r and R being the semi-parameters of the sections of the paraboloid 
on the planes of xz, yz. Now the equation of a normal section of 
this paraboloid, by a plane whose inclination to that of xz is op, will be 
obtained by substituting in this equation x' cos. <p for x> x' sin. <p for ?/, 
z remaining the same for all normal sections (Anal. Geom. ) ; hence, 
the equation of the section in question is 

cos.> sim^ _ fl _ 2R>- 



2r 2R R cos.' 

24 



186 THE DIFFERENTIAL CALCULUS. 

so that the semi-parameter, and, consequently, the radius of curvature 
(94) of this parabolic or hyperbolic section, is 

Rr 

R cos. 2 <p ± r sin. 2 <p' 
the very same as the radius of curvature of the corresponding section 
of the proposed surface, be this what it may (154). Hence, this pa- 
raboloid has the same curvature in every direction that the proposed 
surface has at the origin of the coordinates. 

PROBLEM II. 

(163.) To determine the radius of curvature at any point in an ob- 
lique section. 

Take the tangent to the section through the point as axis of a?, the 
point itself for the origin, and the axis of z in the plane of the section ; 
then, calling the normal the axis of z, the normal section through the 
axis of #, s, and the oblique section a', we have, at the proposed point 

(86), (ds) = (ds'). Now at the proposed point y = J , but if 

\dz ) 

the axis of z' be transferred to the axis of z, then 

z « z > cos. 6 .-. (d 2 z) = (dV) cos. 6; 

hence, by substitution, 

(ds) 2 
y = ■—-? cos. 6 — p cos. 6 . . . . (1), 

(<T2) r V ' 

where y is the radius of the oblique section, and p the radius of the 
normal section through the tangent to the former ; so that y is the pro- 
jection ofp on the plane of the oblique section, which remarkable pro- 
perty is the theorem of Meusnier. 

It immediately follows from this theorem, that, if with the radius of 
any normal section of a curve surface a sphere be described, and 
through the tangent to that section at the normal point planes be drawn, 
cutting both the sphere and the proposed surface, every section of the 
sphere will be an osculating circle to the corresponding section of the 
surface, because, if the normal radius of the sphere be projected on 
any of these sections, the projection will obviously be the radius of 
that section, and the same projection is, by the above theorem, the 
radius of curvature of the corresponding section of the proposed sur- 
face. 



THE DIFFERENTIAL CALCULUS. 187 

Lines of Curvature and Radii of Spherical Curvature. 

(164.) In speaking of plane curves we have already explained (104) 
what is to be understood by consecutive normals and consecutive curves. 
We propose, in the present article, to consider the intersections of 
any normal at a point of a curve surface with its consecutive normal ; 
but here it must be remarked that consecutive normals to curve sur- 
faces do not necessarily intersect, as in plane curves, for, before coin- 
ciding, these normals, although ever so close, need not be both in the 
same plane ; and, in such a case, when they become consecutive, or 
coincide, they coincide throughout at once, having even then no point 
in common that before coinciding was a point of intersection. Hence 
such consecutive normals have no point of intersection. If, however, 
upon any curve surface there can be traced a line, such that the nor- 
mal to the surface at every point of it is intersected by the consecu- 
tive normal, that line will have peculiar properties. Such a line is 
called, by JWonge, a line of curvature. 

PROBLEM III. 

(165.) To determine the lines of curvature through any point on a 
curve surface. 

Let the surface be referred to any rectangular axes whatever, then 
(#', if, «',) being any point on it, we have, for the- equations of the 
normal, 

(A) *-*'+/ (2-o=:(n 

(B)' y-y> + q >(z-z')=0f W' 

Let now the independent variables #', y', take any increments h, k, 
the equations of the normal to the corresponding point will be 
. . dA , . dA 

A + ah rh +lu Tk + ikc ' 

ax ay \ (o\ 

ax ay 

Now, if the normals (1), (2) intersect, their equations must exist 
simultaneously ; therefore, since A = 0, B = 0, 

dA dA k 

dx' dy' h * 

dB ,dB h . ' _ 

dx' dy' h 




188 THE DIFFERENTIAL CALCULUS. 

The coordinates (#, y, z,) of the intersection of the proposed normals 
will be obtained by the combination of the four equations (1) and (3) 
in terms of x', y' z', which are fixed, and of the increments Ar, h. 
But from four equations three unknowns may be always eliminated, 
and the result of this elimination will be an equation between the 
other quantities ; hence then there exists a constant relation between 
the increments k, /i, when the normals intersect, these increments are 
therefore dependent ; consequently the y, x, of which these are the 
increments, must be dependent ;* therefore when the normals are 
consecutive, that is, when h — 0, the equations (3) become 



dA dA dy' 

dx' dy 1 ' dx' 

dB dB 

dx' dy' 



l' dx' ( 



or, by substituting for A and B their values (1), 

1 + P' (p' + 9' %) + V—W + ' %) = o • • • • (*). 

from which, eliminating z' — z% we have the following equation for 
determining -j- 7 

((1 + ^) 8 > - pW g! + ((1 + t) r <- (i + p « )n $L _ 
(1 + p' 2 )s l +p'qV = . . . (6). 
This being a quadratic equation furnishes two values for ~ the 

tangent of the inclination of the projection of the line of curvature 
through (x', ij, z'), on the plane of xy to the axis of x. Hence, there 
are two directions in which lines of curvature can be drawn through 
any proposed point, and if in (6) we substitute for p', q', &c. their 
general values in functions of x, y, that equation will then be the dif- 
ferential equation which belongs to the projections of every pair of 

* If this should appear doubtful to the student, its truth may be shown by re- 
moving the axes of x, y, to the proposed point, in which position k, h, will be the 
variable coordinates of the line of curvature, and these will merely take a constant 
when the axes are replaced in their first position. 



THE DIFFERENTIAL CALCULUS. 



189 



lines of curvature ; so that every line on a curve surface which at all 
its points satisfies this equation, will be a line of curvature. 

(166.) Between every pair of lines of curvature there exists a very 
remarkable relation : it is that they are always at right angles to each 
other. To prove this it will only be necessary to place the coordi- 
nate planes, which have hitherto been arbitrary, so that the plane of 
xy may coincide with, or at least be parallel to, the tangent plane at 
the point to be considered, in which case p' and q' are both 0, and, 
consequently, the equation (6) becomes 

dii 
therefore, calling the two roots or values of —-, tan. <p and tan. cp\ we 

ax 

have, by the theory of equations, 

tan. 6 tan. 6' = — 1,* 

which proves that the projections of the two lines of curvature through 

the origin, are perpendicular to each other, and consequently the lines 

themselves are perpendicular to each other. 

Moreover, the equation (7), if divided by-^j = tan. 2 6 becomes 

identical to equation (6), page 183, which determines the inclinations 
of the principal sections ; hence, the lines of curvature through any 
point, always touch the sections of greatest and least curvature at that 
point. Also, in the same hypothesis, with respect to the disposition 
of the coordinate planes z' = 0, therefore the equation (4) or (5) 
gives 

1 tan. 6 

g — : qy — 

r' + s tan. d s' + t! tan. d' 

out if the plane of xz coincide with a plane of principal section, it 
will, as we have just seen, touch the line of curvature, and then = 0, 
so that 

1 1 

r t 

and these are precisely the expressions found at (152), for the two 
radii of curvature of the principal sections at the proposed point, in 

* Since tangent <p and tangent <p' are the roots of equation, (7), and — 1 is 
their product, recollecting that tangent X cot. = radius 2 = 1, whence y is the 
complement of <t>. Ed. 



190 THE DIFFERENTIAL CALCULUS. 

reference to the same axes ; hence we infer, (161), that the conse- 
cutive normals to the surface at any point, intersect at the same points 
as the consecutive normals to the principal sections. These points 
of intersection, are no other than the centres of curvature of the sur- 
face at the proposed point, for if spheres be described from these 
centres to pass through the proposed point, they will touch there, 
since both have the same normal, and therefore the same tangent 
plane ; and these two spheres have the same curvature as the surface 
in the two directions of the lines of curvature, since consecutive nor- 
mals to the surface in these directions, cut that through the point at 
the centres of these spheres, also the plane sections, tangential to 
these directions, have the corresponding sections of the spheres for 
their osculating circles, since the consecutive normals, at their point 
of contact, also intersect at these centres ; therefore, the radii of cur- 
vature of the surface at any point, coincides entirely with the radii of 
curvature of the principal sections through that point, so that (155) 
if the radii are both equal at any point, the curvature of the surface is 
uniform all round, that point. 

(167.) The annexed figure is intended to give 
an idea of the disposition of the lines of curvature 
on the surface (S), drawn through points P, P', &c. 
PT, P'T', &c. are the normals to the surface at 
those points, and as each is intersected by its con- 
secutive normal, the locus TT' . . . of these in- 
tersections is a curve. The locus too of the nor- 
mals PT, P'T', &c. themselves form a surface, 
throughout perpendicular to the proposed; this 
surface, thus generated by the motion of a straight 
line PT along the curve PF . . . and each position intersecting its 
consecutive position, is obviously a developable surface ; one of 
whose edges is the line of curvature P P' . . . and the other the line of 
centres TT' . . . which latter is called the edge of regression of the 
developable surface. Proceeding, in like manner, along the other 
line of curvature through P, we have another developable normal 
surface, whose edge of regression is the locus of the centres of cur- 
vature belonging to this second line of curvature. Applying similar 
considerations to every point on the surface (S), we shall thus have 
an infinite number of developable normal surfaces at right angles to 




THE DIFFERENTIAL CALCULUS. 191 

each other, and which will obviously form together two continuous 
volumes, and the edges of regression will, in like manner form two 
continuous surfaces, or sheets, being the locus of all the centres of 
curvature. These surfaces, therefore, bear the same relation to the 
original surface, as that which in plane curves we have called the 
evolute bears to the involute. 

It would be quite incompatible with the pretensions of this little 
volume to extend any further our inquiries into the properties of lines 
of curvature. For more detailed information respecting these re- 
markable lines, the student must study the illustrious author by whom 
they were first considered, Monge, in his Application de l y Analyse & 
la Geometrie, a work abounding with the most profound and beautiful 
speculations on the subject of curve surfaces and curves of double 
curvature, and which, together with the Developpements de Geometrie 
of Dupin, constitute a complete body of information on a very at- 
tractive and important branch of mathematical study, the cultivation 
of which, however, has been almost entirely neglected hitherto in this 
country.* 

Radii of Spherical Curvature. 

(168.) We have already seen that the radii of spherical curvature, 
or simply the radii of curvature at any point of a surface, are identi- 
cal to the radii of the principal sections through that point, and have 
given tolerably commodious formulas for the calculation of these radii 
when the axes to which the surface is referred originate at the proposed 
point, the plane of xij being coincident with the tangent plane, and 
the axis of z with the normal at that point. We have also seen that 
when these radii are determined, a paraboloid may also be determined, 
having its vertex at the proposed point and its curvature in all direc- 
tions round that point and in its immediate vicinity, the same as the 
curvature of the surface ; so that be the surface ever so complicated, 
its curvature at any particular point will be correctly presented to us 
by the vertex of a determinable paraboloid. All this, however, sup- 
poses the radii of curvature of the surface at this point to be known ; 

* The only English Mathematician, I believe, who has produced public proof 
of his having given much attention to these inquiries, is Mr. Davies of Bath 
whose papers on surfaces, &c. in Leybourri's Repository, I have already had occa- 
sion to refer to. 



192 THE DIFFERENTIAL CALCULUS. 

it remains* therefore, to 6how how these radii may be determined, 
whatever be the position of the coordinate axes. 

PROBLEM iv. 

(169.) Given the coordinates of a point on a curve surface to de- 
termine the radii of curvature at that point. 

Let (x, y, z,) be the point on the surface, and (x\ y', z',) either of 
the sought points on the normal corresponding to the centres of cur- 
vature, then the radius R from either will be given by the expression 

R 2 = (a? < _ X f + { y> _ yf + {Z > _ Z )«. 

Since at the proposed point the equations (1) and (3), at art. (162), 
must exist simultaneously with this, we have, by substituting in this 
the values of (x' — x), (if — y) as given by (1), 



R = (z' — z)sf l+p' 2 + q'\ 

dv 
Now, if from (4), (5) we ehminate the unknown-^-, we have 

(XX 

(z — z') 2 (r 1 1' — s' 2 ) + (z — z') I (1 + q' 2 ) r' — 2p' q' a' -f 
(l+p /2 )^ + (l+p' 2 +<f) = 0, 
or, putting according to Monge 
g = r ' i! — s' 2 

h = (1 + q') r' — 2p' q's' + (1 + p' 2 ) t> 
P= 1 + p /2 + q' 2 

the equation for determining z — z' becomes 

(*_,>)« + * («_*) + -J = (1), 

and the roots of this substituted in the equation 
R = ( Z _ z ') fc, 

give 



R= -(h±Vh 2 -4 g V) (2) 



2¥ 



W. 



h ± Vh 2 — 4gk 9 
(170.) Thus the radii of curvature are determined, and the direc- 
tions of the lines of curvature, and therefore also of the principal sec- 



THE DIFFERENTIAL CALCULUS. 193 

tions are determined by Problem III. ; consequently, the radius of cur- 
vature of an oblique section, any how inclined to coordinate planes, 
any how situated with respect to the surface, may now be determined 
by help of the formulas (9) and (1) at pages 181 and 184. It ap- 
pears from (3) that the surface will be convex or concave in the di- 
rection of a line of curvature in the immediate vicinity of the point, 
according as g y or g /_ 0. If g = the equation (2) shows that 
one of the radii will be infinite. 

When the functions of x, y, z, represented by p r , q r , r', s', t', are 
complicated, the expressions just deduced for the radii of curvature 
will obviously be complicated in the extreme. They are, however, 
easily manageable when the proposed surface is of the second order, 
as Dupin has shown in his Developpements for both classes of these 
surfaces. We shall here give the solution for surfaces which have 
not a centre, that is for paraboloids ; the process for the other class, 
or for central surfaces, being exactly the same but rather longer. 

PROBLEM V. 

(171.) To determine the radii of curvature at any point in a para- 
boloid. 
The general equation of paraboloids being 

* = f + 2z = 0, . 
A 2 B 2 ' 

we have 

p' = -x>9' = -i--- 1 +*>' 2 + 9' 3 = ^ + ! + 1= * 2 

•••ft--(l+X) F -(l+ S3 ) x jg . 

Hence, generally, whatever be the paraboloid, we have, for the co- 
efficients in equation (1) above, the values 

^ = _A + B- 2 *,|=AB(-l + !- 2+ l)> 
25 



*94 THE DIFFERENTIAL CALCULUS, 

and for R we have 



^ I x 2 v 2 

R = v /_ + L_ + i 

A 2 B 2 
A + B 2z , A + B 2z 2 . n x 2 



The sum of the two radii are, therefore, 



R + r ==y— + i + 1 x (A + B - 2*) 

but (124) the first of these factors is the reciprocal of the cosine of 
the inclination a of the normal at the point (x, y, z,) to the axis of z, 

.-. (R + r) cos. a = A + B — 2z, 
which is the expression for the sum of the projections of the radii of 
curvature on the axis ofz; A, B being the semi-parameters of the 
sections on the planes of xy, yz. If the point be at the vertex, then 
x = 0, y = 0, z = 0, and the values of R then become 



.\ R = A, r = B, 

and these are also the radii of curvature of the two parabolic sections 
on the planes of xy, yz (94), so that these sections which we have al- 
ready called the principal sections in the Analytical Geometry, are 
really the principal sections, or those of greatest and least curvature. 
A similar process leads to similar inferences for central surfaces of 
the second order. 



CHAPTER IV. 

ON TWISTED SURFACES.* 

(172). We have already stated (148) a twisted surface to be one 
whose generatrix is a straight line moving in such a manner along its 
directrices that it continually changes the plane of its motion. 

* This is the class of surfaces called by the French Surfaces Gauches, and 
which, together with the class of developable surfaces, they include under the 



THE DIFFERENTIAL CALCULUS. 195 

The present chapter will be devoted to the consideration of this 
class of surfaces. Proceeding from the simpler kinds to the more 
general, we shall first examine the surfaces whose directrices are 
straight lines as well as the generatrices, then those having one of its 
directrices a curve, afterwards those having two curvilinear directrices, 
and lastly those having three directrices of any kind. 

Tivisted Surfaces having Rectilinear Directrices only. 

PROBLEM I. 

(173.) To determine the surfaces generated by a straight line mo- 
ving parallel to a fixed plane, and along two rectilinear directrices not 
situated in one plane. 

Let the fixed plane, called the directing plane, be taken for that of 
xiji and the plane parallel to the two directrices for that of xz ; then 
the equations of these directrices will be 

/n v x = az + a ) j ( x = ol'z -f- a' /oN 

and the generatrix being parallel to the plane of xij will be represented 
by the equations 

z = b,y = mx + n . . . . (3). 
As this line has always a point in common with (1), the four equations 
(1), (3) exist together, therefore, eliminating x, y, z, we have, among 
the variable- parameters, the relation 

(3 = m (ab + a) + n . . . . (4), 
the parameters a, a, /3, being fixed by the position of the directrices, 
but the others variable. 

In like manner, since the line (3) has also always a point in com- 
mon with (2), the four equations (2), (3) exist together, therefore, 
eliminating x, y, z, we get for a second relation among the three ar- 
bitrary parameters the equation 

general name of Surfaces R6gle"es, expressive of their mode of generation by 
straight line generatrices. There has just appeared, in Leyboum's Repository, 
No. 22, a very masterly inquiry into the history of these surfaces, from the pen of 
Mr. Davies, wherein the claims of the English to the first consideration of "rule 
surfaces " is fully established. 



196 



THE DIFFERENTIAL CALCULUS. 



/3' = m (a'b + a') + n . . . . (5). 
By means of the two relations (4) and (5) among the parameters 
which enter (3), we may eliminate them and thus obtain the sought 
equation in a?, y, z. Subtracting each from (3), we have 

y — /3 — m (x — az — a) 
y — (3' = m (x — a'z — a'), 
eliminating m we obtain, finally, 

(a —a') yz + ( a — a') y + (a'j3 — a/3) z + (&' _ /3) x 

= a<3' — a'jS.... (6) 

for the equation of the surface, which is therefore of the second order. 
Let us now inquire what particular kind of surfaces of the second or- 
der this equation includes. By applying the criteria (3) [Anal. Geom.) 
we find that the surfaces are not central, they must, therefore, be pa- 
raboloids. By putting x = k we find in the resulting equation for 
any section parallel to the plane of yz, that the squares of the variables 
are absent, therefore, (Jlnal Geom.) these sections are all hyperbolas. 
We infer, therefore, that the surface (6) is always a hyperbolic para- 
boloid. If, in the equation (6) we make z equal to any constant 
quantity, the equation will always be that of a straight line, being in- 
deed necessarily one of the positions of the generatrix ; also, if we put 
y equal to any constant quantity, we find that every section parallel 
to the plane of xz is a straight line, so that through every point on the 
surface of a hyperbolic paraboloid there may be drawn two straight 
lines, their assemblage constituting two distinct series situated in 
two distinct series of parallel planes, and hence there are two distinct 
ways in which the surface may be generated by the motion of a straight 
line, but not more than two ways, since the equation (6) represents a 
straight line only on the two hypotheses assumed above ; and as no 
two of the positions of the same generatrix, however close, can be in 
the same plane, the hyperbolic paraboloid is a twisted surface. 

(174.) We may show at once by setting out with the equation of 
the hyperbolic paraboloid, that two straight lines pass through every 
point on its surface, and, moreover, that these lines are both in the 
tangent plane at that point. Thus the equation of the surface is (Jlnal. 
Geom.) 

pa? — p'y 2 ^pp'z .... (1), 



THE DIFFERENTIAL CALCULUS. 197 

and that of the tangent plane through (a?', y\ z',) 

2pxx — 2p'yy' = pp' (z + z') . . . . (2), 
the relation among the coordinates x, y f , z', of the point of contact 
being of course 

px' 2 — p'y' 2 = pp'z' .... (3). 
Adding together equations (1) and (3) and subtracting (2) from the 
sum, there results 

p(x — x') 2 -P f (y — y') 2 = o, 

which is the condition necessary to be satisfied for every projected 
point (a 1 , y,) common to the surface (1) and the plane (2), seeing that 
it has resulted from the combination of their equations. Such con- 
dition being satisfied by every point in the lines represented by the 
equation 

y - if = ± (X - X') y/^, 

it follows that the lines of which these are the projections are common 
to both surface and tangent plane, so that the tangent plane cuts the 
surface according to two straight lines passing through the point of 
contact. 

PROBLEM II. 

(175.) To determine the surface generated by the motion of a 
straight line along three others fixed in position, so that no two of 
them are in the same plane. 

Let us first consider the case in which the three directrices are all 
parallel to the same plane. 

Assume the axes of x and y in this plane passing through one of 
the directrices (B), and parallel to the other two (B'), (B"). Let the 
axis of a? coincide with (B), and the axis of?/ be parallel to (B'), and 
let the axis of z be drawn to pass through both (B') and (B"), then 
the equations of the directrices will be 

(B) y = 0, z = 

(B') x = 0, z = h 

(B") y = ax, z = k 

and the equation of the generatrix in any position will be 

x = mz + p, y = nz + q . . . . ( 1 ). 



198 THE DIFFERENTIAL CALCULUS. 

As this line has always a point in common with the directrices, all 
these equations exist together. Hence, eliminating x, y, z, we have, 
among the variable parameters m, n, p, q, the relations 

q = 0, rah + p — 0, nk = a (mk + p) . . . . (2). 
Eliminating the variable parameters from (1) and (2), we have 
a (k — k) xz == ky (z — h) 

for the equation of the surface sought, and which we find, by applying 
the same tests as in last problem, to be the same surface, viz. the hy- 
perbolic paraboloid. 

(176.) Suppose, now, that the three directrices are not all parallel 
to the same plane, then, taking any point in space for the origin, and 
parallels to the directrices for axes, the equations of these will be 

(B) x — a, y = (3 

(B') z = y, x = a 

(B") y =(3',z = y 

and the equation of the generatrix will be 

x = mz + p, y = nz + q . . . . (1), 
which, since it has a point in common with (B), gives rise to the con- 
dition 

m n 

and having, at the same time, a point in common with (B), and an- 
other in common with (B"), we have the additional conditions 

a! = my + p, (3' = ny + q . . . . (3). 
Eliminating now the arbitrary parameters, m, n, p, q, by means of (1), 
and these equations of condition, we shall arrive at the equation of the 
surface. The equations (3) give, in conjunction with (1), 

x — a w — (3' 
m — , n = — 

z — y z — y 

x — a! y — (3' 



p = a' — y , q = (3' — (3'- 

r ' z — y M 

which values, substituted in (2), give 



z — y " z — y 



(r — /) w + C/ 3 ' — P)xz + (* — *)yz ) 

+ a/3'/ — a(3y' ) 



THE DIFFERENTIAL CALCULUS. 199 

for the equation of the surface. By applying the usual criteria, 
(Anal. Geom.) we find that the surface must be a hyperboloid, and 
as the squares of the variables are all absent from the equation, no 
intersection {Anal. Geom. ) can possibly be an imaginary curve ; hence 
the surface must be a hyperboloid of a single sheet, and it is obviously 
twisted, since the generatrix constantly changes the plane of its 
motion. 

(177.) We may, as in the preceding problem, by commencing 
with the equation of this surface, show that through every point on it 
two straight lines may be drawn, and that they will both be in the 
tangent plane through the point. Thus the equation of the surface 
is 

4+&-4=i — (i). 

a 2 b 2 c 2 
and that of the tangent plane through (#', y', z) 

™ , yy' , zz ' _ -. r2 x 

1? + F + V ~ 1 (2) ' 

the relation among x', y', z', being fixed by the equation 

T t2 ,/2 Z I2 

Adding together equations (1) and (3), and subtracting twice equa- 
tion (2) from the result, we have 

~cT~^ P ? l — m ' { h 

a relation which must have place for every point common to both the 
surface and the tangent plane. 
Also, subtracting (3) from (2) 

*' (* — x ') . y' (y — y') z ' ( z — z ') _ n ,-. 

Now, in order to ascertain whether the points fulfilling these condi- 
tions can lie in a straight line, let us combine them with the equations 
of a straight line through (#', y\ z',) viz. 

x _ x ' = a ' (z — 2'), y—, y ' = b , {z^-z / ) . . . . (6). 
Substituting in the equations (4) and (5) these expressions for x — #', 
y — y', we have, 



200 THE DIFFERENTIAL CALCULUS. 

, ' a'x' , b'v' z' 



a 7 6' 2 J_ = 

a 2 6 2 c 2 

"^ 2 ~ + "6 r ~^~ 
these relations, therefore, must exist among the constants in (6), for 
it to be possible for that line to belong to the surface. From the 
second of these we readily deduce a rational value of a' which, sub- 
stituted in the first, b' will be given by the solution of the quadratic, 
which will furnish two values, so that two lines passing through the 
point of contact maybe drawn, that shall be common to both the 
surface and the tangent plane. 

Twisted Surfaces having but one Curvilinear Directrix. 

(178.) In surfaces of this kind the generatrix moves along a straight 
line and a curve, remaining constantly parallel to a fixed plane called 
the directing plane. Such surfaces are called conoids, and that they 
are twisted surfaces is plain, because a plane to pass through two 
positions of the generatrix must pass through the rectilinear directrix, 
and become, therefore, fixed, so that it cannot be moved round one 
position without ceasing to pass through two. The directing plane 
is usually taken for that of xy, the origin being at the point where the 
straight directrix pierces it. 

PROBLEM III. 

(179.) To determine the general equation of canoidal surfaces : 
Let the equations of the straight directrix be 
x = mz, y = nz . . . . (1), 
and those of the curvilinear directrix, 

F(x,y,z) = 0,/(*, y,z) ■= .... (2). 
The equation of the generatrix, being in every position parallel to the 
plane of xy y must always be of the form 



THE DIFFERENTIAL CALCULUS. 201 

z = a, y — (3x + y . . . . (3), 
a. and (3 being variable parameters. 

As this line has always a point in common with the line (1), their 
equations exist together ; hence, eliminating x, y, z, by means of 
these four equations, we have the condition 

wot = /3ma + y or y = na — (3mu, 
so that the equations (3) of the generatrix become 

z = a,y — na = /3 (x — ma) .... (4) ; 
but this same line has also a point in common with the curve (2) ; 
hence, eliminating x, y, z, by means of the four equations (2), (4), 
we have an equation containing only constants and the variable para- 
meters a, /3, which equation, solved for a, gives 

a = 9 : /3 . . . . (5) * 
But, by equations (4), 

y — nz 

a = z, (3 = Z ; 

x — mz 

hence, by substitution in (5), 

v — nz. 

2 = 9 (- ), 

x — mz J 

which expresses the general relation among the coordinates of any 
point of the generatrix in any position, therefore this is the general 
equation of a conoidal surface. 

(180.) If the straight directrix coincide with the axis of z, then 
m = 0, n = 0, and the conoid is represented by the general equation 

* = <? (|). 

x 
whether the axis of z, or the straight directrix, be perpendicular to the 
directing plane or not; if it is perpendicular, the conoid is called a 
right conoid. In these cases the equations of the generatrix are 
simply 

z = a, y = (3x. 
(181.) As an example, let it be required to find the equation of the 
inferior surface of a winding staircase, the aperture or column round 
which it winds being cylindrical. 

* : is the same as <pP or a function of 0, Ed. I 

26 



202 THE DIFFERENTIAL CALCULUS. 

To conceive the generation of this surface, let us suppose a rect- 
angle to be rolled round a vertical column, which it just embraces* 
the line which was the diagonal of the rectangle will then become a 
winding curve called a helix, and it will make just one turn round the 
column, its horizontal projection being a circle ; if immediately above 
this another equal rectangle be applied to the column, the vertical 
edges when brought together being in a line with those of the first, 
the diagonal of this will form a continuation of the helix, and in this 
way will be exhibited the trace of the edge of the surface in question 
on the vertical column, or the curvilinear directrix ; the other direc- 
trix is the axis of the cylinder, the directing plane being horizontal. 

Now for every point in the diagonal of a rectangle the abscissa has 
a constant ratio to the ordinate, the axes being the sides including the 
diagonal, so that, reckoning from the foot of the helix, the circular 
abscissas and vertical ordinates corresponding are in a constant ratio. 
Hence, taking the centre of the cylindrical base for the origin and 
drawing the axis of y through the foot of the helix, calling h the 
height and 2irr the base of one of the rectangles, or of the cylinder, 
we shall have for each point of the helix, these relations, viz. 

x 2 + if = r 2 , x — r sin. — , — = - — .... (1), 
J r s 2<kt 

and for the generating line the equations 

z = a, y = fix . . . . (2). 

If from the two last of (1) we eliminate the arc s we shall have the 

following equations of the projections of the curve 

2irz 
x + f — r ^ x = r sin. (-7-) .... (3), 

eliminating x, y, z, from the equations (2), (3) we have 

in which equation if we substitute for a and (3 the values z and — 
given by (2), we shall obtain, finally, 

"" Z X z 

= sin. (2?r T ) or - = tan. {2* T ) 



s/x* + f h y h 

which is the equation of the surface, that is of the twisted helixoid. 



THE DIFFERENTIAL CALCULUS. 203 

(182. ) It remains to determine the differential equation of conoidal 
surfaces. In order to this we must eliminate the arbitrary function 
<p in the equation 

y — nz 
T K x — mz 
by differentiating, as in the several similar cases in Chapter II., we 
thus obtain the equation 

p' _ p' ( m y — nx ) — (^ — nz ) 

q' q' (my — nx) + (x — mz) 
which reduces to 

p' ( x —. mz) -f q' (y — nz) — 0, 

or when thelconoid is right simply to 

p'x + q'y = 0, 

because then m = 0, n = 0. 

(183.) The same results may be at once obtained from the con- 
sideration of the tangent plane ; for (x r , y', z\) being any point on the 
surface, the equation of the tangent plane is 

z — z = p (x — x') + q (y — y'), 
which touches the surface along the generatrix through (x, y', z'), 
and this being every where at the same distance z' from the horizon- 
tal plane, it follows that if in the above equation we put z — z' the re- 
sult 

p'{x—x) -f q' [y — y') = 
will express the relation between the x, y of every point in this gene- 
ratrix. But at that point where it cuts the straight directrix, the a?, y 
have the relation 

x — mz', # = fts'? 
so that, by substitution, we have 

p' (mz — x') ■+• q' (nz' — y') = 0, 

for the relation among the coordinates of every point (a?', y', z',) on the 
surface,' which agrees with that deduced above. 

Twisted Surfaces having Curvilinear Directrices only. 
(184.) We now proceed to consider those surfaces which cannot 



204 THE DIFFERENTIAL CALCULUS. 

have a rectilinear directrix, or rather those whose directrices may be 
any lines whatever. We shall first suppose two directrices. 

PROBLEM IV. 

(185.) To determine the general equation of surfaces generated 
by a straight line which moves along any two directrices (D), (D') 
whatever, and continues at the same time parallel to a fixed plane. 

Taking as before the directing plane for that of xy, the equation of 
the generatrix in any position will be 

* = a » y — P x + 7 • • • • (i)» 

the parameters all varying with the varying positions of the generatrix. 
Let now the equations of the two fixed directrices be 

(D) F 0, y, z) = Q,f{x, y, z) = . . . . (2) 

(DO F 1 (*, y, z) = 0,/, (*, y, *) = . . . '. (3). 
Then the condition is, first that the generatrix meets (D), or that 
their equations (1), (2) exist together; hence by eliminating the co- 
ordinates of the common point from these four equations, we shall 
obviously obtain an equation containing only constants and the varia- 
ble parameters a, (3, y, that is to say, we shall obtain among these 
parameters a relation 

.<*> (a, /3, 7 ,) = 0. 

Proceeding in the same manner with the equations (1), (3) which 
also exist together for a certain point, we obtain a second relation 

Y (a, 13, y,) = 0. 

By means of these two equations we may eliminate any one of the 
parameters ; therefore, eliminating first y and then j3, we have 

(3 = (p:a,y — -^:a; 

hence, substituting for these variable parameters their values in func- 
tions of the variable coordinates as furnished by equation (1), we 
have, for the general relation among these coordinates, the equation 

xy — cp:z + 4 ,:2 • • • • (4). 

This then is the general equation of all surfaces generated as an- 
nounced, whatever be the form of the directrices ; when these forms 
are given, the forms of p and 4, become determinable by the above 



THE DIFFERENTIAL CALCULUS. 205 

process, and then the general equation (4) takes the particular form 
belonging to the individual surface. 

(186.) Let us now determine the general equation of these surfa- 
ces in terms of the partial differential coefficients. Putting the equa- 
tion (4) in the form 

y — xcp:z = -^/iz, 

the ratio of the partial coefficients of each side, taken relatively to x 
and y, will, by the principle in (58), be 

— <p:2 p p' 

—~— = —„ that is, ^-7 = — cp:z, 
I q' q 

an equation from which the arbitrary function -^:z is eliminated. Ap- 
plying the same principle to this last equation we have 



dx dy q' 

that is, putting according to the usual notation 

dx dx dy ' dy 

qY — p's' . q's — p't' _ p 



q' 2 q' 2 q 



whence 



9 'V — 2p'q's' + p'H' - 0, 

an equation from which both the arbitrary functions are eliminated, 
and which must be fulfilled for every point in every surface generated 
as in the problem, whatever be the directrices. We see that as two 
arbitrary functions were to be eliminated, the process led to a partial 
differential equation of the second order. 

PROBLEM V. 

(187.) To determine the general equation of surfaces generated 
by the motion of a straight line along three curvilinear directrices (D) 
(DO, (D"). 

We shall first remark that the motion of the generatrix is entirely 
governed by these conditions, for if we take any point on the first di- 
rectrix (D) and conceive two cones whose bases are (D')» (D") to 



206 THE DIFFERENTIAL CALCULUS. 

have this point for their common vertex, these cones will obviously 
intersect each other in all the straight lines that can be drawn from 
the point to the curves (DO, (D"), the positions of these lines are 
therefore fixed by these intersecting cones, and these are fixed by their 
bases ; hence, all the lines that can be drawn from the point to the 
lines (D'), (D'O are determinate both in number and position, this 
being true for every point in (D), it follows that the surface generated 
by all these lines is determinate, and it is now required to find its 
equation. 

As there is here no directing plane the equations of the generatrix 
in any position will take the form 

x = az + y, y = j3z + 5 . . . . (1), 
and, since it always has a point in common with (D), we may elimi- 
nate by means of the equation of (D) combined with these, the coor- 
dinates of that point : the result will furnish a condition among the 
variable parameters. In like manner, employing the equation of 
(D') we shall arrive at another equation of condition, and, lastly, the 
equation of (D'O will furnish a third equation. By means of these 
three equations any two of the parameters a, (3, y, S, may be elimi- 
nated, and we shall obtain three equations of the form 

(3 = cp:a, y = -^:u, 6 = i?:a. 

substituting these expressions for (3, y, 8, in the equations (1) we have 

x = az + -vj,:a, y = Z(p:a, + ir:a, 

two equations which have place for every surface generated as pro- 
posed, the functions which fix the directrices being quite arbitrary. If 
these functions are known, or the directrices fixed, we may then eli- 
minate the parameter a by means of these equations, and thus deduce 
the equation of the individual surface, but the general relations among 
the coordinates for all the surfaces of this family can be exhibited only 
by means of two equations as above. The general relation among 
the partial differential coefficients belonging to all this family of sur- 
faces may, however, be ascertained in a single equation by eliminat- 
ing, as in last problem, all the arbitrary functions by successive dif- 
ferentiation ; this will lead to a partial differential equation of the third 
order, for which see JWonge's Application de V Analyse a la Geometrie^ 
p. 195. 



THE DIFFERENTIAL CALCULUS. 



207 




(188.) We shall terminate the present chapter with the following 
example : 

On the opposite sides of the hori- 
zontal parallelogram AB DC are de- 
scribed two vertical semicircles, and 
perpendicular to their planes is drawn 
the straight line OY through the centre 
of the parallelogram ; taking this 
straight line and the two semicircles 
as directrices, it is required to find 
the equation of the surface generated 
by a straight line moving along them. 

Let the axes of coordinates be the 
perpendicular horizontal lines OX, 
OY, and the vertical OZ, then the equations of the three directrices 
will be 

z = 0,2 = .... (1) 

y = — 6, (x — a) 2 + z 2 = r 2 . . . . (2) 
y = + b, (x + a) 2 + z 2 = r 2 . • . . (3). 
The equations of the generatrix, since it always passes through a 
point (/•?, 0, 0) in the axis of?/, will take the forms 

x = a(y — (3),z = r (y — (3) . . . . (4), 
and the condition to be fulfilled by this line is, that it rests on each of 
the semicircles ; or that at certain points, x, y, z, are the same in the 
equations (2), (4) and (3), (4) ; hence, eliminating these first from 
(2), (4), and then from (3), (4), we have these relations among the 
variable parameters, viz. 

\*(b + /3) + a \ 2 + f(b + (B) 2 = ^ 
\a (6 + /3) + a\ 2 + f (b — (3) 2 = r 2 
which, by subtraction, give 

/3 (ba 2 + aa + bj3 2 ) = 0. 
This condition is satisfied by the value (3 = 0, but this is not admis- 
sible, since it would restrict the generatrix to pass always through 

the origin, and have no motion along OY ; hence, dividing by — , 



. (5) 
• (6), 



208 THE DIFFERENTIAL CALCULUS. 

we have the relation 

« a + f + f = (7), 

between the parameters a, y. 

Substituting the value of y 2 , given by this equation in (5), it be- 
comes 

(6 2 _ (3") au = b (r 2 — a 2 ) ... . (8), 
and by means of these equations, together with those of the genera- 
trix, we may readily eliminate the parameters ; thus the values of a 
and 7, given by (4), are 

x z 

a = , y = — , 

and these, substituted in (7), give 

, b(a* + z 2 ) ax 2 



8 



ax 6(* 2 + 2 2 )' 

and finally, these substituted in (8) give for the surface the equation 

which is the same as 

\axy + b {x 2 + a 2 ) | a = b 3 r 2 x 2 +b 2 {r 2 — a 2 ) z 2 . 



CHAPTER V. 

ON DEVELOPABLE SURFACES AND ENVELOPES. 

(189.) When in an equation between three variables 
F (x, y, z, a) = 0, 
there enters an arbitrary constant a, that equation, by giving different 
values to a, will represent so many different surfaces all belonging to 
the same family. If we fix one of these by any determinate value of 
a, another, intersecting this, will be represented by changing a into 
a + h, h being some finite value. If h be now continually dimi- 
nished, the intersection will continually vary, and will become fixed 



THE DIFFERENTIAL CALCULUS. 



209 



only when the varying surface becomes coincident with the fixed 
surface. In this position the intersection is said to belong to con- 
secutive surfaces, and it may be determined both in form and position 
by a process similar to that employed at (105). Thus a being the 
only variable concerned in the intersections, let u = F (r, y, z, a), 
now if a increase by h, u = F (x, y, z,a + h) which developed by 
Taylor's theorem, gives 

du d 2 u hr 

u = u 4. h + TT . — -f &c. - 

da da? 2 

but, since u = 0, therefore 

du . d 2 u h 

-r + tt • o + &c * = ° ; 

da da A 2 
hence, when the surfaces are consecutive, that is, when h — 0, we 
have the following equations for determining the curve of intersection, 

viz. 

du = \ 

du = \ . . . . (1), 

Ik y 

these, therefore, are the equations of the curve, which is the intersec- 
tion of the surface (1) with its consecutive surface. 

If from these two equations we eliminate a, the result will be the 
general relation among the coordinates of every point in every such 
consecutive intersection throughout the whole family of surfaces, this 
resulting equation will therefore represent the surface which is the 
locus of all these consecutive intersections. This locus, moreover, 
touches each of the variable surfaces throughout their intersections ; 
for differentiating the equation F = of any one of the variable sur- 
faces, a being constant, we have 

du du f _ du du , _ 

dx dz ' dy dz 

and, differentiating the equation F = of the locus, a being variable, 
we have 

du du du da n du , du . . du da. 

-j— + -7- P +-J— •-7- ==0 '-T- + T-9+rr- •^- = ' 
ax dz x da dx dy dz T da dy 

which equations are identical to those above, since 

du 



da 



27 



210 



THE DIFFERENTIAL CALCULUS. 



and therefore each pair give the same values for p' and q consequently, 
at the points common to both surfaces they have common tangent 
planes. Hence the locus of the consecutive intersections touches, 
and envelopes all the variable surfaces ; it is, therefore, called by 
Monge the Envelope of these surfaces. 

(190.) If the envelope be formed by the consecutive intersections 
of planes, then, since from what has been just proved, the envelope is 
touched throughout each of the intersections by the corresponding 
plane ; this envelope is such that the tangent plane at any point 
touches it throughout, the rectilinear generatrix passing through that 
point ; and this is the characteristic property of a developable surface . 
hence a developable surface may be considered as the envelope of a 
family of planes represented by the general equation 

z — Ax + % + D . . . ,. (1), 
in which there enters a variable parameter a. 

Now, to introduce this variable parameter in the most general 
manner possible into the equation (1), we ought to consider each of 
the coefficients A, B, C, to be functions of it, so that the general form 
will be 

z —foe + xcpa -j- ycpoi,* 

and therefore the line of contact (189) or generatrix of the surface 
will be represented by the equations 

z =f a + X (pa + ij^a ) . 

=f'a-\-xcp'a + y-\>'a J " * " " V ;• 

When the forms of the functions/, <p, -j,, are fixed, the variable para- 
meter a may be eliminated, and the resulting equation in x, y, z, will 
be that of the individual surface to which these particular forms be- 
long. The equations (2), therefore, may be considered as represent- 

* Monge says the general equation may always be put under the form 
z = x<pa -f- yipa -f- a, 
which, however, seems to be incorrect, since it excludes those of the family com- 
prehended in the equationf 

z = X(pa -{- yipa -\- c, 
and which evidently generate conical surfaces, whose vertices are all on the axis 
of z, at the distance c, from the origin. The form in the text includes this class 
of equations, for fa may be constant without causing (pa or ^a to become so. 

] Monge in this observation is not incorrect, since a is indeterminate, one of its 
values will necessarily be equal to c. Ed. 



THE DIFFERENTIAL CALCULUS. 



211 



ing the whole family of these surfaces, a in the first being a function 
of x and y, implied in the second. 

(191.) 1 . As an example, suppose it were required to determine 
the developable surface generated by the intersection of normal planes 
at every point in a curve of double curvature. 

Representing the proposed curve by the equations 
y' = Fx\z'=fx' .... (1), 
the general equation of the normal plane will be (129) 

— ' + W (r-y)+#(^-')=o.....(2), 

in which the only variable parameter is x ; y and z being determinate 
functions of it given by the equations of the curve. Hence, differ- 
entiating with respect to x', we have 

fJr.,'2 f ]„>2 J2„ f J2~, 

Now the functions of x', which enter the equations (2), (3), being 
given by (1), we may eliminate this parameter from them, and the 
resulting equation in x, y, z, will be that of the developable surface 
required. 

(192.) 2. As a second example, let it be required to determine 
the developable surface which touches and embraces two given curve 
surfaces. 

If we suppose one of these surfaces to be a luminous surface en- 
lightening the other, the surface which we seek will obviously em- 
brace all the rays which proceed from the bright surface to the dark 
one, and the curve of contact on this latter, will separate the illuminated 
and dark parts. 

Let the equations of the two given surfaces be 

Fi (*i. y» *r) = °> F 2 (»» 2fc> %). = • • • (1). 
then the equations to tangent planes to each will have the form 

z — z l9 = Pl — x x ) + q { (y — t/,) . . . (2) 
and 

z — z-2 = vz ( x — **) + & (y — y»)i 

and for these planes to belong to both surfaces, their equations must 
be identical, that is, we must have the conditions 

Pi =-Jto <?i ~q-2 . . • (3), 






212 



THE DIFFERENTIAL CALCULUS. 



*i ~ PM — qiy* = % —}hx> — qjy-2 • • • (4). 
By means of the six equations marked, five of the coordinates may 
be determined in terms of the sixth, x ; hence, if these functions of x, 
be now substituted for their values in the remaining equation, we shall 
obtain a result containing only the variable parameter x } , and which 
will consequently represent the family of planes which generate the 
developable surface sought. Calling this result P = 0, the genera- 
trix of the surface will be given by the equations 

P = 0,f = 0, 

dx x 

from which, eliminating x n we have the equation of the surface 
sought ; and this equation, combined with that of each surface sepa- 
rately, will give the two curves of contact. 

PROBLEM. 

(193.) To determine the differential equation of developable sur- 
faces in general. 

The general equation of the generating plane, arranged according 
to the variable coordinates x, y, z 9 of any point in it, is 

z = p'x + q'y + z — p'x — q'y' ... (1) 
and this plane remains the same for every point in the generatrix, as 
well as for the point (#',' y\ z f ), so that the quantities 

p',q',z'—p'x' — q'y' . . . (2), 
remain constant, although x\ y\ z', all vary, provided this variation is 
confined to the rectilinear generatrix, for which y is always a function 
of x, but not else ; hence, the conditions which restrict the point x\ 
y', z\ to the generatrix on which it is first assumed, is, that the dif- 
ferential coefficients derived from (2), y being considered as a func- 
tion of x, are all 0, and it is plain that if any two be 0, the third will 
be also ; hence, differentiating the two first, we have 

ax ax 

dii 
where ~^- fixes the position of the rectilinear directrix for which the 
ttx 

dv 
expressions (2), remain constant. Eliminating, then — , we obtain 

the following equation, which must hold for every directrix, viz. 



THE DIFFERENTIAL CALCULUS. 213 

r'f — s' 2 = . . . (-3). 

This, therefore, is the equation which the differential of the equa- 
tion of every developable surface must accord with ; or, in usual 
terms, it is the differential equation of developable surfaces in general. 

(194.) We shall exhibit another method of obtaining the equation 
(3), from the general equations (2), art. (190). Differentiating the 
first of these, in which a is a function of x and y implied in the se- 
cond, we have the two partial differential equations 

/ r, dec da. ... da 

. da , da . , da 

but, in virtue of the second of the equations (2), these become 

p' = (pa, q' = 4>a, 
consequently, p> must be a function of </', and may therefore be re- 
presented by 

p' = irtf. 
Eliminating now the arbitrary function it by differentiation, as in 
(58), we have 

s t 
as before. 

(195.) We have as yet considered only the simplest class of sur- 
faces, whose intersections, with their consecutive surfaces, are given 
by the general equations (1), art. (189), viz. plane surfaces, the in- 
tersections being straight lines. It is obvious, however, that what- 
ever be the surfaces, the intersections are still given by the equations 
(1), and the envelope of these surfaces, found by eliminating from 
them the arbitrary parameter a. This parameter, however, may 
enter the equation of any particular family of surfaces in an infinite 
variety of different forms and ways ; it may enter into only one of its 
terms, or be combined with several ; a simple power only of it may 
enter, or a complicated function, and still, entering only as a parame- 
ter, the general equation, under all these changes, will still preserve 
the same character, and represent but one family of surfaces. With 
the envelope, however, it will be different ; this depends as well on 



214 THE DIFFERENTIAL CALCULUS. 

the arbitrary parameter, as on the variables which enter the general 
equation, since the value of this parameter must be found from one 
of the equations (1), in terms of x, y, z, and this value substituted in 
the other for the equation of the envelope. Nevertheless, since, as 
just observed, the individual surfaces represented by the two equa- 
tions (1), for every particular value of the parameter in whatever form 
it may enter, is always of the same degree, it follows that each indi- 
vidual intersection, (1), will uniformly be a curve of the same order, 
and which will change its order only when the order of the surface 
changes. This curve of intersection or of contact, common to all 
the envelopes of the same family of surfaces, is called, by JWonge, 
the characteristic. 

Considering any of the characteristics (1) separately, we may in- 
quire what are the points in which it is intersected by the consecutive 
characteristic ; and the method of determining these intersections is 
analogous to that already explained in (105) and (189), that is, we 
must combine with the equations (1) of this curve their differentials 
taken relatively to a • hence, the consecutive intersections for any 
particular position of the characteristic, will be determined by the 
equations 

u = 

do? ~ U ' 
each of these separately represent a surface, any two together a line 
common to both, and all three the point or points common to their 
intersection, a being considered constant. By solving these three 
equations for x, y, and z, we shall obviously obtain known values for 
the coordinates of the points of intersection required, which of course 
are all situated on the envelope. 

Now, if from the three equations above, we eliminate a, we shall 
have two equations in x, y, z, existing together, which, being the same 
for the intersections of every pair of consecutive characteristics must 
represent the locus of these intersections, and be situated on the en- 
velope. It will therefore be a line which touches and encompasses 
all the characteristics, in the same manner as the envelope touches 



THE DIFFERENTIAL CALCULUS. 215 

and embraces all the enveloped surfaces. It must then form an edge 
of the envelope, or the line in which its sheets terminate, and it is 
therefore called, hy JMonge, the edge of regression of the envelope. 
In the developable surfaces, we have seen that the characteristic is a 
straight line, and the consecutive intersections of the characteristic, 
in every position, obviously form the edge which limits the locus of 
the characteristics, that is, the developable surface. 

The consideration of envelopes, characteristics, and edges of re- 
gression, have been successfully employed by JMonge, and succeed- 
ing writers, to remove several difficulties in the higher departments 
of the integral calculus, that do not appear to be otherwise clearly ex- 
plicable ; but it would be out of place here to more than to hint 
at the importance of these researches ; to pursue them to their fullest 
extent the advanced student must have recourse to the profound 
work of JMonge, before referred to, viz. Application de V Analyse 
a la Geometric. We shall conclude the present chapter, with one ex- 
ample on the determination of the envelope. 

(196.) The centre of a sphere of given radius moves along a given 
plane curve, it is required to determine the surface which envelopes 
the sphere in every position. 

Let the equation of the given curve, along which the centre moves, 
be 

P = <pa . , . . (1), 
so that for every abscissa a of this curve, the ordinate corresponding 
will be (pa; therefore, the variable coordinates of the centre of the 
sphere are a, (pa ; hence its equation, in any position, is 

(ar _ a) 2 + (y — cpaf + a" = 'r* . . . . (2), 
hence the equations of the characteristic are, 

(*—a)»+(y — (p a )» + *» = ,*} 

x — a + (y — (pa ) (p'a = ) * * * * ^ '" 

The last equation is that of a plane, passing through the point (a, 
(pa), or centre of the sphere ; it is, moreover, perpendicular to the tan- 
gent to the curve (1) at this point, for the equation of this tangent is 

(j3 r — (3) = (p'a ((pa' — <pa), 
and that above is 

j,_ <p a = — (a__ a ) f 



216 THE DIFFERENTIAL CALCULUS. 

so that whatever be the form of <p, the characteristic is always a great 
circle of the moveable sphere, of which the plane is normal to the 
curve. The species of the curve which is the characteristic, being, 
however, constant, as observed in art. (195,) however 9 and a may 
vary, the species may be at once determined by assuming a = 0, 
<pa == 0, which reduces the equations of the characteristic to 

x 2 + f + z 2 
x = 

which belongs to a circle ; the species, therefore, is a curve of the 
second order. 

To determine the equation of the envelope, we must eliminate a 
from (3), and the resulting equation in x, ?/, z, will belong to the en- 
velope ; thus, if the curve (1) be a circle of radius a, then 

— 7 — a 

(pet — \/ a 2 — a 2 . *. (p'a 



-}- 



V a 2 — a 2 
substituting these values in the equations (3), they become 



(* —a) 2 + (y — Va 2 —a 2 ) + z 2 = r\ 



ay = x \f a — a 2 , 
and determining, from this last equation, the expression for a, and 
substituting it in the preceding, we shall obtain, finally, 



(a± Vx 2 + if) =r* — z 2 , 
for the equation of the envelope. 



CHAPTER VI. 

ON CURVES OF DOUBLE CURVATURE. 

(197.) In the preliminary chapter to the present section, we inves- 
tigated the expressions for the tangent lines and normal planes to these 
curves ; we shall now discuss their general theory. As, however, in 
the course of this discussion, we shall sometimes have occasion to 
employ the differential expression for the arc of a curve of double 



THE DIFFERENTIAL CALCULUS. 217 

curvature, we shall commence by seeking the form of this expres- 
sion. 

(198.) We know that the projecting surface of every curve of dou- 
ble curvature, is a cylindrical surface, {see Anal. Geom.) if, therefore, 
this cylindrical surface be developed, the curve will become plane, 
and its length will be unaltered, and the curvilinear base of the project- 
ing cylinder, which we shall here suppose to be vertical, will become 
a straight line on the plane of xy ; hence, for the plane curve referred 
to this straight line t, and the axis of z, we shall have (86) the ex- 
pression 

(ds) 2 = (dz) 2 + (dt)\ 
but t being itself in reality the arc of a plane curve, we have 

{dty = (dxf -f (c%) 2 , 

hence, by substitution, 

(ds) 2 = (dx) 2 + (dyf + (dz) 2 , 
which is the differential expression required. 

Osculation of Curves of Double Curvature. 

(199.) Let 

y =fx, z = Fx . . . . (1), 
and 

Y = ^x, Z = Tx , . . . (2), 

be the equations of two curves of double curvature, or rather of the 
projections of these curves on the planes of xy, xz : then, if we con- 
sider the constants «, b, c, &c. which enter the first pair of equations 
as known, and the constants A, B, C, &c. belonging to the second 
pair as arbitrary, these latter may be determined so that the curve to 
which the)i,belong may touch the proposed or fixed curve (1), in any 
given point, more intimately than any other curve of the family (2). 
For, giving to x any increment, h, we have, by Taylor's theorem, 

and it has been shown, (87), that if the constants which enter the first 

28 



218 THE DIFFERENTIAL CALCULUS. 

of the equations (2), be determined, all of them from the conditions 

„ dy dY dhj d 2 Y _ 

^•s^s^^a?-* (3) ' 

the projection of the curve (2), on the plane of xy, shall touch, more 
intimately, the projection of (1) on that plane, than the projection of 
any other curve of the family (2). 

In like manner, if the constants which enter the second of the equa- 
tions (2), be all of them determined from the conditions 

_ dz _ dZ drz _ d 2 Z 

' dx dx' dx 2 dx 2 ' 

the projection of the curve (2) on the plane of xz, will touch more in- 
timately the projection of (1) on that plane, than the projection of any 
other curve of the family (2). 

It is clear, therefore, that if all the constants in the equations (2), 
be determined conformably to the conditions (3), (4), the curve (2) 
will touch more intimately the curve (1 in space, than any other curve 
of the family (2), and that the contact will be the less intimate as the 
conditions (3), (4), satisfied by the arbitrary constants, are fewer. 

The conditions for simple contact, or contact of the first order, are 
evidently 

= Y,z = Z, d *=— , — = — . (5), 

' dx dx dx dx 

and, for contact of the second order, we must have the two additional 
conditions 

d 2 y _ d?Y_ tfz_ _ d?Z_ 
~d? d^F' ~dx~ 2 d?"' 

and so on. 

(200.) From these principles, we may very easily 'deduce the 
equations of the tangent at any point of a curve of double curvature. 

Thus the equations of any straight line in space, are 

y = Ax + B ; z = A'x + B' . . . . (1), 

and these correspond to the equations (2) above, and as four arbi- 
trary constants enter, the conditions (5) may be fulfilled by them ; 
thus, taking the two last conditions, we have, by accenting the varia- 
bles of the curve, 



THE DIFFERENTIAL CALCULUS. 219 

and, therefore, the two first require that 

so that the equations (1) of the tangent, through any point (x 1 , y', z'), 
are, 



. dz ' , * 

Z Z — -— [X — X ) 

ax ' 



(2). 



PROELEM I. 

(201.) To determine the osculating circle, at any point in a curve 
of double curvature. 

In finding the osculating circle, at any point of a plane curve, we 
had of course, the plane of that curve given, but, in the present case, 
we have to determine both the plane of the circle, and its radius. 
Now let us suppose that r is the radius of the osculating circle, and 
a » fit /? are the coordinates of its centre, then it is plain, that the 
circle will be a great circle of the sphere whose equation is 

(a: — a ) 2 + {y — [3f + (z — yf = r 2 . . . . (1), 

and since the plane of the circle passes through the point (a, /3, y,) its 
equation must be of the form 

a?-_ a + m (y — £) + n (z — c) — . . . . (2). 
These two equations, combined with those of the proposed curve, 
give the values of x, y, z, common to all, and therefore belong to the 
point where the circle (1), (2), meets the curve. We have, therefore, 
to differentiate the equations (1), (2), successively, and to consider, 
agreeably to the conditions (3), (4), art. (199), that the resulting dif- 
ferential coefficients belong as well to the proposed curve at the 
point, as to this circle. For contact of the first order we have 
( x -a) + (y-(3)p' + ( Z -c)q' = .... (3), 
1 -f- mp' + nq' = . . . . (4), 
and for contact of the second order we have, in addition, 



220 THE DIFFERENTIAL CALCULUS. 

1 + p> 2 + q> 2 + p" (y - (3) + q" (z-y) = .... (5), 

mp" + nq" = .... (6). 
All these six conditions, therefore, must exist for the contact at the 
point (a?, y, z,) in the proposed curve to be of the second order; and 
as the equations (1), (2), of the touching curve, contain six disposable 
constants, viz. a, /3, y, r, m, n, all these conditions may be fulfilled, 
but no more ; hence, the circle, determined agreeably to these con- 
ditions, will touch the proposed curve more intimately than any other, 
that is, it will be the osculating circle. From equations (4) and (6) 
we get 

q p" 

q'p —p'q' q 'p"—p'q" 

hence, equation (2) becomes 

x — a + -r-^ — (y — (3) — - r - s £ — (z — 7 )= 0, 

qp —pq y qp —pq 

or 

hence, the three conditions (2), (4), (6), determine the plane of the 
osculating circle, and which is called the osculating plane, through 
the proposed point (x, y, z.) Equation (7) then represents this 
plane. 

For the coordinates of the centre of the osculating circle we have, 
from equations (1), (2), (3), 

{up — mq) r (n — a') r 

*-« = — M ,y-.P j^— . 

(m — p')r 

Z — y = =— — , 

where M is put for the expression 

V \(np' — mq') 2 + (» — q) 2 + (m — p'Yl- 
Substituting these values in (5) we have, for the radius of the oscu- 
lating circle, 

(1 -f p' 2 + q 2 ) M 
( n _ 9 ')p"_ ( w __ p') q "' 

Hence, putting for m and n the values already deduced, and restoring 
the value of M, we have 



THE DIFFERENTIAL CALCULUS. 221 



r = _... (l+^ + g 2 ) 1 



Vlp" 2 +q" 2 +(p'q"-qY) 2 V 

p" 2 + g" 2 + (p'g" — qp"Y J 

8 - « -L ( X + ?' 3 + g' 2 ) lp" — P (pfr" + g'g ") { 
p - + q 2 + (pq —qp) 

y = * + ^ + P' 2 + 9 /2 ) fa" — 9 (Pi>" + g'g ") \ 
p" 2 + g" 2 + [ p 'q —qff 

(202.) The expression for r may be rendered more general, by 
considering the independent variable as arbitrary ; in which case we 
have (66), 

= (d 2 y) (dx) — (d 2 x) (dy) „ = (d 2 z) (dx) — ( d 2 x) (dz) 
V {dx) 3 ' q {dxf 

Also (198) 

hence, making these substitutions in the above expression, we have 

{dsf 

"" V \ {dx) (d*y)-(dy) {dH) \ 2 +{dz){d*x)-{dx) (d*z) | 2 +(^) (d*z)— (dz) (d*y) \ 2 \ 

(203.) If it were required to determine the circle having contact 
of the first order, merely with the proposed curve, only the conditions 
(1), (2), (3), (4), must be satisfied; the conditions (2), (4), deter- 
mine the plane of this circle, that is the tangent plane, but as the 
condition (4) leaves one of the constants m, n, arbitrary, the tangent 
plane is not fixed, but may take an infinite variety of positions ; but 
as it must necessarily pass through the linear tangent, which is fixed, 
it follows that a plane through this, and revolving round it, is a tan- 
gent plane in every position, in one of which it touches the curve 
with a contact of the second order, and thus becomes the osculating 
plane. 

(204.) There is another method of determining the equation of 
the osculating plane, very generally employed by French authors ; 
they consider a curve of double curvature to have, at every point, two 
consecutive elements, or infinitely small contiguous arcs in the same 
plane, but not more, the plane of these elements being the osculating 



222 THE DIFFERENTIAL CALCULUS. 

plane at the point. The process, then, is to assume the equation of 
a plane through the point 

x — x' + m (y — y') + n (z — z') = (1), 

and to subject it to the condition of passing also through the points 

(a? + dx', y' + dy', z' + dz*), 
and 

x' + 2dx' + d 2 x, y' + 2dy' + d 2 y\ z' + 2dz + dV. 
Such a process, the student will at once perceive to be exceedingly 
exceptionable ; for besides the vague notion attached to the infinitely 
small consecutive arcs, the expressions x -f- dx, y + dy, and the like, 
mean no more in the language of the differe7itial calculus, than x, y, 
&c, for dx, dy, &c. are not infinitely small, but absolutely 0, as we 
have all along been careful to impress on the mind of the student. 
The process is, however, susceptible of improvement thus : suppose 
the plane (1) passing through one point (x', y', z') of the curve passes 
also through a second point, of which the abscissa is x' + A x', where 
Ax' means the increment of x, then substituting x + A x' for x, the 
equation (1) becomes 

x — x + »* (y — y') + n {z — z') — (Ax + mAy' + nAz') 
= . . (2), 
which, in virtue of (1), is the same as 

Ax' -\- mAy' + nAz' = 0, 



Ay' , Az 
Ax' Ax' 

Suppose now that these two points merge into one, that is, let 
Arc' = 0, then 

dy' , dz' 

hence the plane becomes determinable by the conditions (1), (3). 

Again, let this plane pass through a third point, x -f- Ax', then sub- 
stituting this for x in both the equations (1), (3), they will furnish the 
additional condition 



dy' dz' 



THE DIFFERENTIAL CALCULUS. 223 

hence, dividing by Ax', and supposing this third point to coincide with 
the former, that is, supposing Ax' = 0, we have the new condition 

„_jL + ._=0 (4). 

The equations (3) and (4), determine m and n, and thence the 
plane (1), which is such as to pass through but one point of the curve T 
and at the same time to be so placed that the most minute variation 
from this position will cause it to pass through three points of the 
curve. 

(205.) By whatever process the osculating plane is determined, 
the radius of the osculating circle may be easily found from consider- 
ations different from those at (201). For, as the linear tangent to 
the curve, must also be tangent to the osculating circle, it follows 
that the centre of this circle must be on the normal plane, as well as 
on the osculating plane ; it must, therefore, lie in the line of intersec- 
tion of this normal plane, with its consecutive normal plane ; hence, 
if this line be determined, the combination of its equation with that 
of the osculating plane, will give the point sought. Now (189) the 
line of intersection of consecutive normal planes is 

x — x' + p' (y — y') + q >(z-z>) = ) 

v" (y-yl + q" (? — *) -p' 2 - </' 2 - 1 = o f 

therefore, the centre is to be determined by combining these equa- 
tions with that of the osculating plane, viz. 

being precisely the same equations as those employed before, for the 
same purpose. If the origin be at the point, and the tangent be the 
axis of x, then x\ y', z', p', q', are each ; therefore, the equations of 
the line of intersection are 

q" , 1 

x = 0, y = — lyjz +— , 
p" p 

and the equation of the osculating plane 

p" Z - q "y = 0; 

this, therefore, is perpendicular to the line of intersection. (Anal. 
Geom.) 

(206.) The expressions in (201) for the coordinates of the centre 



224 THE DIFFERENTIAL CALCULUS. 

of the osculating circle will become very simple by introducing the 
substitutions furnished by art. (202); the results of these substitutions 
will be 

d 2 x _ , „ dry „ d 2 z 



* = x + r 2 ~£n P = y + r*-y-, 7 =: Z +r> 



ds 2 ' ** lJ l ' ds 2 ' r ~ * ' ' ds 2 ' 
the independent variable being s. (See JVoie D.) 

PROBLEM II. 

(207.) To determine the centre and radius of spherical curvature 
at any point in a curve of double curvature. 

We are here required to determine a sphere in contact with the 
proposed curve at a given point, such that a line on its surface in the 
direction of the proposed may in the vicinity of the point be closer to 
the curve than if any other sphere were employed. In the direction 
of the curve the z and the y of the sphere must be both functions of 
x, so that the equation of the sphere is resolvable into two, corres- 
ponding to the equations (2) art. (199), which two equations belong 
to the curve which osculates the proposed. The actual resolution of 
the equation into two is obviously unnecessary ; it will be sufficient 
in that equation to consider x as the only independent variable. 

The general equation of a sphere is 

(aj _ a) 2 + (y-(3) 2 + (z - y) 2 = r 2 . . . . (1), 
and the particular sphere required will be that whose constants are 
determined from the following differential equations : 

x — a+p'Xy — (3) + q'(z — y) =0 .... (2) 

p"{y — P) + q" ( z — r) + l + p >2 + q 2==0 • • • • ( 3 ) 
p'" (y - P) + 4" {* - 7) + 3 W + q'q") - ° W- 

These four equations fix the values of the parameters a, (3, y 9 r, 
and, therefore, determine both the position and magnitude of the os- 
culating sphere. If the origin of coordinates be at the proposed 
point, and the linear tangent be taken for the axis of x, the determina- 
tion becomes easy, for x, y, z, being each = 0, as also p\ q\ the fore- 
going equations (2), (3), (4), become 

a = 0,p"/3 + q'y = l,/"/S + f'y = 0, 

7 = —-^—; 



PT —Pq qp'—q'p 



THE DIFFERENTIAL CALCULUS. 225 

iience, by substitution in (1), 



V „'"> + 



p'q" — q'p 
(208. ) We already know that if to every point in a curve of double 
curvature normal planes be drawn, the intersections of these planes 
with the consecutive normal planes will be the characteristics of the 
developable surface which they generate, and the intersection of any 
characteristic with the consecutive characteristic will be a point in 
the edge of regression, corresponding to the given point on the pro- 
posed curve. Now equation (2) above being that of the normal 
plane, this point is determined by precisely the same equations (2), 
(3), (4), as determine the centre of spherical curvature, these points, 
therefore, are one and the same, as might be expected ; hence the 
locus of the centres of spherical curvature forms the edge of regres- 
sion of the developable surface generated by the intei sections of the 
consecutive normals. If then by means of one of the equations of 
the proposed curve and the three equations of condition mentioned 
we eliminate x, y, z, and then perform the same elimination by means 
of the other equation of the curve and the same conditions, we shall 
obtain two resulting equations in a, (3, y, which will be the equations 
of the edge of regression. 

PROBLEM III. 

(209.) To determine the points of inflexion in a curve of double 
curvature. 

Since a curve of double curvature as its name implies has curva- 
ture in two directions, if at any point its curvature in one direction 
changes from concave to convex the point is called a point of simple 
inflexion. But if at the same point there is also a like change of 
curvature in the other direction, the point is then said to be one of 
double inflexion. In other words, if but one projection of the tan- 
gent crosses the projected curve the point is one of simple inflexion, 
but if the tangent cross the curve in both projections then the point is 
one of double inflexion. As in plane curves the tangent line has 
contact one degree higher at a point of inflexion, so here the contact 
of the osculating plane is one degree higher. Hence, at such a 
point besides the conditions in (201) which fix the osculating plane, 

29 



226 THE DIFFERENTIAL CALCULUS. 

we must at a point of simple inflexion have the additional condition 
arising from differentiating (6), viz. 

mp"' -f nq" = 0. 
Eliminating — from this and equation (5) we have 

pV = gY". 

which condition renders the expression for the radius of spherical 
curvature at the point infinite, as it ought.* Unless, therefore, this 
condition exist, the point cannot be one of inflexion; but the point 
for which the condition holds may be one of inflexion, yet to deter- 
mine this the curve must be examined in the vicinity of the point. 

As to points of double inflection, it is evident from what has 
been said (121) with respect to plane curves that such points must 
fulfil the conditions 

p'' = or go , q" = or oo , 
and these render the radius r of absolute curvature infinite or 0. 

Evohites of Curves of Double Curvature. 

(210.) In opeaking of the evolutes of plane curves we observed 
(103,) that the evolute of any plane curve was such that if a string 

* The French mathematicians consider a point of simple inflexion to be that at 
which three consecutive elements of the curve lie in the same plane. In a recent 
publication from the university of Cambridge the author has attempted to deduce 
the above equation of condition, by viewing the point of inflexion after the manner 
of the French. He has however confounded the consecutive elements of a curve 
with what the same writers term consecutive points ; moreover, after having estab- 
lished the conditions necessary for the plane 

z — Ax + By + C, 
passing through one point (x, y, z,) in the curve, to pass also through two points 
consecutive to this, viz. the conditions 

dx ax ' ax dx 

where y„ z n belong to one of the consecutive points, it is inferred that 
_f£_ = B ^/_ d*z, _ B <Py, 
dx* dx" 1 ' dx* dx* 

an inference which is quite unwarrantable, and which cannot exist unless the plane 
pass through/owr consecutive points instead of three. 



THE DIFFERENTIAL CALCULUS, 227 

were wrapped round it and continued in the direction of its tangent 
till it reached a point in the involute curve, the unwinding of this string 
would cause its extremity to describe the involute. But besides the 
plane evolute hitherto considered, there are numberless curves of 
double curvature round which the string might be wound and con- 
tinued in the direction of a tangent till it reached the involute, which 
would equally, by unwinding, describe this involute ; and generally 
every curve, whether plane or of double curvature, has an infinite 
number of evolutes, as we are about now to show. 

(211.) If through the centre of a circle, and perpendicular to its 
plane, an indefinite straight line be drawn, and any point whatever 
be taken in this line, then it is obvious that this point will be equally 
distant from every point in the circumference of the circle, so that, if 
a line be drawn from it to the circumference, this line, in revolving 
round the perpendicular under the same angle, will describe the cir- 
cumference. Such a point is called a. pole of the circle, so that every 
circle has an infinite number of poles, the locus of which is determined 
when the places of any two are given. 

(212.) Now, as respects curves of double curvature, we have seen 
that the centre of the circle of absolute curvature corresponding to 
any point is in the line where the normal at this point is intersected by 
its consecutive normal, the centre itself being that point in this line 
where it pierces the osculating plane, which (205) is the plane drawn 
through the tangent line perpendicular to this line of intersection, or 
characteristic ; hence the characteristic corresponding to any point 
in the curve is the locus of the poles of curvature at that point, and 
the intersection of this characteristic, with the perpendicular to it from 
the corresponding point of the curve, is that particular pole which is 
the centre of absolute curvature, the perpendicular itself being the 
radius. 

As the locus of the poles corresponding to any point is no other 
than the characteristic, the locus of all the poles corresponding to all 
the points of the curve must be the locus of all the characteristics, 
and therefore (190) a developable surface. 

(213.) Suppose now through any point, P, of the curve a normal 
plane is drawn of indefinite extent, the characteristic or line of poles 
corresponding to the point will be in this plane ; let, therefore, any 
straight line be drawn from P to intersect this line of poles in the point 



228 THE DIFFERENTIAL CALCULUS. 

Q, and be continued indefinitely. If this normal plane be conceived 
to move, so that, while P describes the proposed curve, the plane 
continues to be normal, the characteristic will undergo a correspond- 
ing motion, and will generate the developable surface corresponding 
to the curve described by P, and this motion of the characteristic will 
cause a corresponding motion of the point Q, not only in space, but 
along the arbitrary line from P, which has no motion in the moving 
plane. As, therefore, Q moves along the characteristic successive 
portions QQ' of the line, PQ will apply themselves to the surface 
which the moveable characteristic generates, and there form a curve 
to which always the unapplied portion QP is a tangent. Now the 
normal plane being in every position tangent to the surface through- 
out the whole length of the characteristic, it is obvious that, in the 
above generation of this surface, nothing more in effect has been done 
than the bending of the original normal plane, supposed flexible, into 
a developable surface. If, therefore, we now perform the reverse 
operation, that is, if we unbend the normal plane, the point P will de- 
scribe the curve of double curvature, and the curve QQ' traced on 
the developable surface will become the straight line PQ ; so that the 
curve of double curvature may be described by the unwinding of a 
string wrapped about the curve Q'Q, and continued in the direction 
QP of its tangent, till it reaches the point P in the proposed curve. 
It follows, therefore, that the curve Q'Q is an evolute of the curve of 
double curvature proposed, and, moreover, that, as the line PQ ori- 
ginally drawn was quite arbitrary, the proposed curve has an infinite 
number of evolutes situated on the developable surface, ivhich is the 
locus of the poles of the proposed; hence the locus of the poles is the 
locus of the evolutes. 

If the original line PQ be perpendicular to the corresponding line 
of poles or characteristic, then, since this characteristic moves in the 
moving plane while PQ remains fixed, PQ cannot continue to be per- 
pendicular to the characteristic ; but the radius of absolute curvature 
is always perpendicular to the characteristic, this radius therefore 
cannot continue to intersect the characteristic in the point Q, so that 
the locus of the centres of absolute curvature is not one of the evolutes 
of the proposed curve, 

(214.) Should the curve which we have all along considered of 
double curvature be plane, then, indeed, since the characteristics are 



THE DIFFERENTIAL CALCULUS. 229* 

all parallel, and perpendicular to the plane of the curve, the line PQ 
once perpendicular will be always perpendicular to the characteristic, 
so that then Q will coincide with the centre of curvature, PQ being 
no other than the radius of curvature, the locus of the centres being 
the plane evolute before considered. But when PQ is not drawn per- 
pendicular to the original characteristic, but is inclined to it at an an- 
gle a, then it always preserves this inclination during the generation 
of the cylindrical surface which is the locus of the poles, therefore 
every curvilinear evolute of a plane curve is a helix described on the 
surface of the cylinder, which is the locus of the poles of the plane 
curve. 

Every curve traced on the surface of a sphere, has, for the locus of 
its evolutes, a conical surface whose vertex is at the centre of the 
sphere ; because the normal planes to the curve being also normal 
planes to the spheric surface, all pass through the centre. 

(215.) From what has now been said, it is obvious that if from any 
point in a curve a line be drawn to touch the developable surface which 
is the locus of its poles, and its prolongation be wound about the sur- 
face without twisting,* it will trace one of the evolutes, and, as the 
string may be drawn to touch the surface in every possible direction, 
it follows that every developable line on the surface will be an evo- 
lute. If the curve be plane, the evolutes are all on the cylindrical 
surface whose base is the plane evolute. 

As obviously a developable line is the shortest on the surface that 
can join its extremities, it follows that the shortest distance between 
two points of an evolute measured on the surface is the arc of that 
evolute between them. 

PROBLEM IV. 

(216.) Having given the equations of a curve of double curvature 
to determine those of any one of its evolutes. 

All the evolutes of the curve being on the same developable sur- 

* This is what I understand Monge to mean, when he says (App. de PJlnal. de 
Geom, p/ 348,) "si l'on plie librement sur cette surface le prolongement de cette 
tangente." It seems not improper to call such lines placed on a developable sur- 
face developable lines, and those which form curves on the developed surface twist- 
ed lines. Of these two species of lines all the former are evolutes, but none of the 
latter are. 



230 THE DIFFERENTIAL CALCULUS. 

face, the equation of this surface must be common to them all, and 
we have already seen (194) how the equation of the surface is to be 
determined, so that it only remains to find for each evolute a particu- 
lar equation which distinguishes it from all the others, and determines 
its course on the developable surface. In order to this let us consi- 
der that each evolute must be such that the prolongation of its tan- 
gent at any point always cuts the involute, or, which is the same 
thing, the tangent to the projection of the evolute at any point passes 
through the corresponding point in the projection of the evolute ; 
therefore, considering the plane of xy as that of projection, we have, 
for the tangent at any point (a?', y') in the projected evolute, 

and, since the same line passes through a point (x, y,) in the project- 
ed involute, its equation is also 

Y — y' = Sin* (X — x') 

J X X 

dx' x' — x ' 
hence, combining this equation with that of the developable surface, 
determined agreeably to the process pointed out in article, 194, and 
eliminating x, y being a given function of x, we shall have two equa- 
tions in $', y, z't of which one will contain partial differential coeffi- 
cients of the first order, and which together will represent all the evo- 
lutes. To find that particular one which is fixed by any proposed 
condition, it will be necessary to discover, by the aid of the integral 
calculus, the primitive equation from which the differential equation 
mentioned is deducible ; this primitive equation will involve an arbi- 
trary constant, whose value may be fixed by the proposed condition, 
and thus the equations of the particular evolute will be determined. 

We shall terminate this section by subjoining a few miscellaneous 
propositions. 



THE DIFFERENTIAL CALCULUS. 231 



CHAPTER VIZ. 
MISCELLANEOUS PROPOSITIONS. 

PROPOSITION I. 

(217.) To prove that the locus of all the linear tangents at any 
point of a curve surface is necessarily a plane. 

This property we have hitherto assumed ; it may, however, be de- 
monstrated as follows : 

Let the equation of any curve surface be 

i =/(«,») — (i), 

x and y being the independent variables. 

Through any given point on this surface let any curve be traced, 
then, the projection of this curve on the plane of xy will be represented 
by 

y = ?* .. . . . (2), 
which will equally represent the projecting cylinder ; hence the com- 
bination of the equations (1), (2), completely determines the curve, 
and its projection on the plane of xz may be found by eliminating y 
from these equations ; the result of this elimination will be the equa- 
tion 

z=f(x,cpx) =4# . |. . (3), 
therefore, since the linear tangent in space is projected into tangents 
to these two curves (2), (3), its equations must be 
, _ dcpx' 



dx' 
, d-^x 



(x—x') 
(x — x) 



I 



dx' 

where x', y', z', are the coordinates of the proposed point on the sur- 
face. Now -f- is the total differential coefficient derived from the 
dx 

function z = f (x, y), in which y is considered as a function of x given 
by the equation (2), that is 



232 THE DIFFERENTIAL CALCULUS. 

d^x _ dz f ^ f d<px 

~dx~~*dx~l ~ V ' 9 ~dx~' 

hence, by substitution, the equations of the tangent in space become 

(XX 

Now, to obtain the locus of the tangents whatever be the curve through 
the point (#', ij', z'), we must eliminate the function <px, on which alone 
the nature of the curve depends. Executing then this elimination by 
means of the equations (4) and there results for the required locus 
the equation 

z — z > =z p ' (x~x') + q'{y — y'), 
which is that of a plane. 

PROPOSITION II. 

(218.) Given the algebraic equation of a curve surface to deter- 
mine whether or not the surface has a centre. 

That point is called the centre which bisects all the chords drawn 
through it, so that if the equation of the surface is satisfied for any 
constant values x', y', z', it will equally be satisfied for the same val- 
ues taken negatively, that is, for — x', — y', — z', provided the ori- 
gin of coordinates be placed at the centre, so that if no point exists 
for the origin of coordinates, in reference to which the equation 

f(x,y,z,) = 
of the surface remains the same whether the signs of the variables be 
assumed all + or all — , then we may conclude also that no centre 
exists. 

The mode of proceeding, therefore, is to assume the indeterminates 
x t , y t , e j , for the coordinates of the unknown centre, and to transport 
the origin of the axes to that point by substituting in the equation of 
the surface x -f- x t , y + y t , z + z, , for x, y, z. This done we may 
readily deduce equations of condition which will give the proper val- 
ues of x t , y t , z /5 if a centre exists, or will show, by their incongruity, 
that the surface has no centre. Thus, suppose the equation of the 
surface is of an even degree, then we must equate to the coefficients 



THE DIFFERENTIAL CALCULUS. 233 

of all the odd powers and combinations of x, y, z, since the terms into 
which these enter would change signs when the variables change 
signs : we obtain in this way the equations of condition. If the equa- 
tion of the surface be of an odd degree, then we must equate to zero 
the coefficients of all the even powers and combinations of a?, y, z ; so 
that only odd powers and combinations may effectively enter the 
equation, for then whether the variables be all + or all — the function 
f(x, y, z,) will still be 0. 

Now the differential calculus furnishes us at once with the means 
of obtaining the several expressions which we must equate to zero 
without actually substituting x + a?,, y + y t , z-\- z n for x, y, z, in the 
equation of the surface. For if we conceive these substitutions made 
in the function/ (x, y, 2), we may consider the result as arising from 
x a Vs z s taking the respective increments x, y, z, and we know that 
every such function may by Taylor's theorem be developed accord- 
ing to the powers and combinations of the increments, and that the 
several terms of the development consist each of the partial differen- 
tial coefficients of the preceding term, the first being/ (x t , y t , z,). 
Hence, if the coefficients of the first powers of x, y, z, are to be re- 
spectively zero, then we have to equate to zero each of the partial 
coefficients derived from u, = f (#, y t , z ,) = 0, or, which is the same 
thing, from u=f(x, y, z,) — the proposed equation ; if the coeffi- 
cients of the second pow T ers and combinations of x,y, z, are to be ren- 
dered each 0, then we shall have to equate to zero each partial coef- 
ficient derived from again differentiating, and so on. 

As an illustration of this, let the general equation of surfaces of the 
second order 

Ax 2 + Mif + M'z 2 + 2Bt/z + 2Wzx + 2B"xy \_ n _ H 

+ 2Cr + 2Q'y + 2C"« + E ) ~ U ~ u '" (1 ) 

be proposed, then the degree of the equation being even, the coeffi- 
cients of the odd powers of the variables in the equation arising from 
putting x -f- a? # , y + y,, z + z,% for x, y, z, are to be equated to 0, and 
as the equation is but of the second degree, these odd powers will be 
of the first ; hence we have merely to equate the first partial differen- 
tial cofficients to 0, that is 

30 



234 



THE DIFFERENTIAL CALCULUS. 



^ = A* + B'z + B"y + C = 

(in 

T = A'y + Bz + B"* + C = 

^ = M'z + B* + By + C" = 



(2). 



The values of x, y, z, deduced from these equations are the coordi- 
nates x t , y t , z fl of the centre. These values may be represented by 



LP 



N' 



N" 



D'*'~D~ 



where 

D = AB 2 + A'B' 3 + A"B" 2 — AAA" — 2BBB", 
so that the surface has a centre if D is not 0, but if D = and the 
numerators all finite, the surface has no centre, and, lastly, if D = 
and either of the numerators, also 0, then the surface has an infinite 
number of centres, and is, therefore, cylindrical. 

The equations of condition (2) are the same as those at page 

of the Analytical Geometry. 



PROPOSITION III. 

(219.) To determine the equation of the diametral plane in a sur- 
face of the second order which will be conjugate to a given system 
of parallel chords. 

Let the inclinations of the chords to the axes be «, (3, y, then the 
equations of any one will be 

x = mz + p» y — nz + q • • • . (1)» 
where 



cos. a 
m = , n 



cos. (3 



cos. y cos. y 

For the points common to this line and the surface we must combine 
this equation with equation (1) last proposition, and we shall have a 
result of the form 

Rz 2 + Sz + T = . . . . (2), 
which equation will furnish the two values of z corresponding to the 
two extremities of the diameter, and therefore half the sum of these 
values will be the z of the middle, that is, 



THE DIFFERENTIAL CALCULUS. 235 

s = _ § |-...2R S = S + (3), 

which is obviously the differential coefficient derived from (2), or, 
which is the same thing, the total differential coefficient derived from 
(1) last proposition, in which x and y are functions of z given by the 
equations (1). This differential coefficient is, therefore, 

du _ du dx du dy du _ 
dz* dx dz dy dz dz 
du . du . du 

where p and q, the only quantities which vary with the chord, are 
eliminated ; hence, this last equation represents the locus of the mid- 
dle points of the chords or the diametral surface, and it is obviously 
a plane. 

By actually effecting the differentiations indicated in equation (4) 
upon the equation (1) last proposition, we have for the equation of the 
required diametral plane, 

m (A* + B'z -f B"y + C) -f- n (A'y + Bz + Bx + C) 
+ k"z + By + B'x + C" == 0, 
or 

(Am + B' + B"n) x -f (A'n + B + B"m) y + 
(A" + Bn+ B'm) z + Cm -f C'n -f C" = 0. 

PROPOSITION IV. 

(220.) A straight line moves so that three given points in it con- 
stantly rest on the same three rectangular planes ; required the sur- 
face which is the locus of any other point in it. 

Let the proposed planes be taken for those of the coordinates, and 
let the coordinates of the generating point be x, y, z, and the invaria- 
ble distances of this point from the three points resting on the planes 
of yz, xz, and xy, X, Y, Z. The coordinates of these three points 
will be 

In the plane of yz, 0, y\ z' 

xz, x", 0, z" 
xy, x"\ y" 0. 



236 THE DIFFERENTIAL CALCULUS. 

Then, since the parts of any straight line are proportional to their 
projections on any plane, each part having the same inclination to it, 
it follows that if we project successively each of the parts X, Y, Z, 
on the three coordinate planes, we shall have the relations 

x x — x" x — x' :i 



X Y 


Z 


y — y'_ y 


_y-y'" 


X Y 


z 


z — z' __z — z" 


z 



1 



\ : ■ ■ "'• 

X Y Z 

But the part X of the moveable straight line comprised between the 
generating point (x, y, z,) and the point (0, y n 3,,), resting on the plane 
of y, z, has for its length the expression 



(2) 





X 2 = 


or + (y — 


y'f + (z - 


-z 


or 












_ x 2 

1 — ^2 = 


{y — v'f 

" X 2 




• 


bul 


; from the equations 


(1) 








y 


— y' _ y 

X Y 


' X 


z 
Z 


hei 


ice, by substitution, 


(2) becomes 








x 2 y 2 

X 2_r Y 2 







consequently, the surface generated is always of the second order. 
The surface would still be of the second order if the three directing 
planes were oblique instead of rectangular, as is shown by JW. Dupin, 
in his Developpements, p. 342, whence the above solution is taken. 



proposition v. 



(221.) To determine the line of greatest inclination through any 
point on a curve surface. 

The property which distinguishes the line of greatest inclination 
through any point is this, viz. that at every point of it the linear tan- 
gent makes with the horizon a greater angle than any other tangent 
to the surface drawn through the same point of the curve. Now, as 
all the linear tangents through any point are in the tangent plane to 



THE DIFFERENTIAL CALCULUS. 237 

the surface at that point, that one which is perpendicular to the trace 
of the tangent plane will necessarily be the shortest, and therefore 
approach nearest to the perpendicular, that is, it will form a greater 
angle with the horizon than any of the others. We have, therefore, 
to determine the curve to which the linear tangent at every point is 
always perpendicular to the horizontal trace of the tangent plane to 
the surface through the same point, or, which is the same thing, the 
projection of the linear tangent on the plane of xy must be perpen- 
dicular to the trace of the tangent plane. 

Now the equation of the projection of the linear tangent at any 
point is 

and, by putting z = in the equation of the tangent plane, we have 
for the trace in the plane of xy the equation 

— z'=p'{x — x') + q' {y — ij), 
and, since these two lines are to be always perpendicular to each 
other, we must have throughout the curve the general condition. 

dx p' ' ' dx ^ 
p' and q' being derived from the equation of the surface ; so that the 
values of these being obtained in terms of x and y, and substituted in 
the equation just deduced, the result will be the general differential 
equation belonging to the projection of every curve of greatest inclina- 
tion that can be drawn on the proposed surface. To determine that 
passing through a particular point, or subject to a particular condition, 
we must, by help of the integral calculus, determine the general 
primitive equation from which the above is deducible, this primitive 
will involve an arbitrary constant which may be fixed by the proposed 
condition, and thus the particular line be represented. 



PROPOSITION vi. 

(222.) The six edges of any irregular tetraedron or triangular 
pyramid are opposed two by two, and the nearest distance of two op- 
posite edges is called breadth; so that the tetraedron has three 



238 



THE DIFFERENTIAL CALCULUS. 



breadths and four heights. It is required to demonstrate that in every 
tetraedron the sum of the reciprocals of the squares of the breadths is 
equal to the sum of the reciprocals of the squares of the heights. 

Let the vertex of the telraedron be taken for the origin of the rec- 
tangular coordinates, and let also one of the faces coincide with the 
plane of xz, then the coordinates of the three corners of the base will 
be 

0, 0, z', | x", 0, z", | x"\ y'", z", 

and the equations of the three edges terminating in the vertex will be 



x = 
y = 



Now the perpendicular distance between each of these edges and the 
opposite edge of the base will evidently be equal to the perpendicular 
demitted from the origin on a plane drawn through the latter edge? 
and parallel to the former. Hence, denoting the three planes through 
the edges of the base by 

Ax+By + Cz = l\Ex+Fy+Gz=l\Ix+Ky + La = l 9 
they must be drawn so as to fulfil the conditions (See Anal. Geom.) 



x" 
x = — z 


x" 

y'" 
y=^, z - 


y = 



Qz =1 
Ex"+Fy'"+Gz'"=l 

Ex" +Gz" =0 



Qz =1 

Ax" +Cz" =1 

Ax""+By"'+Cz"'=0 

These conditions fix the following values for A, B, C, &c, viz 



Ix" +Lz"=l 

Ix'"+Ky'" + Lz"=l 

liz =0 



1 1 z x"z X 

— A=— — , D = —7T-777-, — 

z x x z x y z x y 

1 z" 1 x"z" 

z x z y x y z 

1 1 t" 

x y x y 



y * 



y z 



Hence, calling the breadths B, B', B", we have (Anal. Geom.) 

— = a 2 + b 2 + c a - (yV ~ y'" z y+( x '" z "— x '" z '— x " z '"Y+ ( x "y'"Y 

B 2 (x"y'"z) 2 

1 „„.,....«., (y , V) 2 + (x"z' + x'"z"—x"z'"y-\- (x"y"f 

(xY'z'f 



B 



= E 2 +F 2 +G 3 = 



THE DIFFERENTIAL CALCULUS, 239 

1 (y'"z') 2 +(x"z — x'"z) 2 

B * (xy z) 2 

Hence ^ + -^- I -g^ 

^WOY^+K^-^V' -^'1 2 + „ I -^(WzY (1). 

j(a'— a">"+»"V'} 2 +(y"V) a +(a;"— #'") 2 * *f ' { J ] K ] 

Again, the expressions for the heights or perpendiculars demitted 

from each of the points 

(0, 0, 0) ; (0, 0, z') ; {x", 0, z") ; (*'", y"\ z'"), 

upon the plane which passes through the other three are, severally, 

(Anal. Geom.) 

u 2 = («YV) 2 

{z"—z'f if' 2 + I {z" — z) x" + (z — z") x'"\ 2 + {y'"x"f 

H , 2 WW 



{z'YJ + {x"z"' — x"'z"f + (x"y"J 

h- = (a/yv)2 h- = {x "y'" z ' )2 

{f'z'f + {x'"z'Y {x"z') 2 

H 2 H 2 H /2 T H 



1112. 



(*'—9W*+ \ (Z'"—Z')X"+ ( Z i—Z")x"'\ 2 + \ 

2{x"y i ") 2 +(y" J z") 2 +(x"z>"— x'"z") 2 +(y"' 2 +x" l2 +x" 2 )z'*) 
-r {x»y'»zj . . . (2), 

which expression is the same as that before deduced, and thus the 
theorem is established by a process purely analytical. This remarka- 
ble property was discovered by M. Brianchon, and formed the sub- 
ject of the prize question in the Ladies' Diary for 1830 : a solution 
upon different principles may be seen in the Diary for 1831. 



END OF THE DIFFERENTIAL CALCULUS, 



NOTES. 



Note {A), page 19. 

^The expressions for the differentials of circular functions are all 
readily derivable, as in the text, from the differential expressions for 
the sine and cosine. We here propose to show how these latter may 
be obtained, independently of the considerations in art. (14). 

By multiplying together the expressions 

cos. A + sin. A V — 1, cos. A Y + sin. A : V — 1, 
the product becomes 

cos. A cos. A 1 — sin. A sin. A 1? 



= (cos. A sin, A x + sin. A cos. A x ) \f — 1. 
But (Lacroix's Trigonometry.) 

cos. A cos A! — sin. A sin. A l = cos. (A + A x ) ) 
cos. A sin. A t + sin. A cos. A x = sin. (A 4- A,) ) ' ' * (■ '' 
Tience the product is 



cos. (A + A,) + sin. (A + A x ) v/— 1, 



= cos. A' + sin. A' V — 1.* 
Consequently, the product of this last expression, and 



cos. A 2 + sin. A 2 V — 1, 



cos. (A' + A 2 ) + sin. (A' + A 2 ) V— l,f 
— cos. A' 7 + sin. A" V — 1, 
the product of this last, and 

cos. A 3 + sin. A 3 \/ — 1, 



cos. A'"-f sin. A" 1 V—l. 

* Writing A' for A + A,, f Writing A" for A' -f A2 &c, Ed, 

31 



242 NOTES. 

Hence, generally, 



(cos. A + sin. A V — l)(cos.Ai-f-sin.Ai\/ — l)(cos.A 2 +sin.A 2 \^— 1) 

&c. = 



cos.(A + A 1 +A a +A3+&c.)sin.(A + A 1 + A a + A 3 +&c.)^— 1. 
Supposing, now, 

A = A, = A 2 = A 3 = &c. 
this equation becomes 

(cos. A + sin. A \f — l) n = cos. wA ± sin. nA V — 1, 
or, since the radical may be taken either -f or — 



(cos. A ± sin. A V — l) n = cos. ?iA ±. sin. nA \f — 1, 
which is the formula of Demoivre, n being any whole number. 



n 
Put a = — A then 
m 



(cos, a ± sin. a \f — l) m = cos. ma ± sin. ma \f — 1, 



= cos. nA ± sin. nA V — 1 = (cos. A ± sin. A \/ — l) n , 
therefore, extracting the mth root, 



n n 



cos. — A dr sin. — A\/ — 1 ■= (cos. A dt sin. A v — l) m 
mi m v ' 

which is the formula when the exponent is fractional. 

Having thus got Demoivre's formula, we may immediately deduce 
from it, as in art. (22), the series 

cos. nA = cos. n A cos. n ~ 2 A sin. 2 A + &c. 

i. • » U ( n l)( Jl 2 ) » -* • 1» i o 

sin. nA=ncos. Asm. A — ^ -cos." "Asm. A+&c. 

Let n = £, sin. A = = A .•. »A = £ = any finite quality #, 
hence, by these substitutions, the foregoing series become 

cos . a ,= 1 __L_ + _^___ &c . 

consequently, 

x 2 x* 

d sin. x = (1 — - — - + - — — &c.) dx = cos. xdx, 

1 . Ji 1 . £ . o • 4 



NOTES. 243 

4 X"* x^ 

a cos. x — — (x + — &ic.)dx = — sin.xdx 

K 1.2.3 1.2.3.4.5 } 

I had intended to have given here another method of arriving at the 
differentials of the sine and cosine, and to which allusion is made at 
page 41, but, upon close examination, I find that the process I had 
then in view is liable to objection, and is therefore best omitted. 



Note (B),page 91. 
Demonstration of the Theorems of Laplace and Lagrange. 

Let it be required to develop the function 

u = Yz where z = F (?/ + xfz). 
By differentiating the second of these equations, first relatively to 
x, and then relatively to v, we have 

Multiplying the first by — and the second by — , and subtracting, 

there results 

dz dz dz dz 

dx-f Z dj- Q '''Tx =fZ dy' * ••W' 

but since u or Yz depends only on s, we shall have 

du dz du , dz 

ax ax ay dy 

therefore, eliminating Y'z, we get 

du dz du dz __ 

dx dy dy dx ' 

dz 
or putting for — its value (1), and making for abridgment/* = Z, 

* At page 89 we put/'z to represent the differential coefficient of/z relatively 
to «; here the same symbol denotes the coefficient relatively to z. 



244 



NOTES, 



dz 
this last expression becomes divisible by -r- and reduces to 

du flu 

so that we may always substitute for — the quantity Z — . 

If we differentiate the preceding equation relatively to x, we shall 
obtain 

du 

dhi <ty r2 x 

= — j .... \i), 

dx 2 dx 

but the expression Z — being no other than/s.Y 'z — , that is to say 

dz 
a function of z multiplied by — , we may consider it as the differential 

coefficict of some new function of 2, which we may represent by u ly 
and we shall then have 

du 

du± du dy d 2 u^ 

= Z ~r-, and 



dy dy 1 dx dxdy 

therefore (2), inverting the order of the differentiations in this last ex- 
pression, 

du x 
d?a d 2 Ui dx 

dx 2 ~ dydx ~ dy ' ' ' ' v '" 
now it must be observed that the relation (A) exists, whatever be the 
function u ; it therefore exists for the function u„ hence 

d u i _ y diit 
dx dy ' 

Substituting then in (3) for -^- its value here exhibited, and after- 
dx 

wards for -^- its equal Z -p-, there results 
dy n dy 

du, du 

- dz -r dz 2 -r- 

d 2 u dy_ dy 

dx 2 dy ~ dy ' ' ' ' K h 



NOTES. 245 



Differentiating this last equation relatively to x, we shall obtain 

d 2 Z 2 — 
d 3 u dy 

dx 3 dxdy ' 

du 
and considering, as before, the function Z 2 -=- to be the differential 

coefficient of some new function of z, viz. u 2i we shall have, by invert- 
ing the order of the differentiations, 

d 2 du 2 
du 2 __ r72 du d 3 u ' dx 

dy dy dx 3 dy 2 • ■• • • V /» 

but the equation (A) subsisting for every function w, must have place 

for the function w 2 , hence 







du 2 _ 
dx 


dy' 




therefore 


(4). 










du 2 
dx 


_- 3 du d 3 u 
dy 1 dx 3 


d 2 Z 3 
dy 


du 
dy 

3 



. . (C). 

The analogy among the expressions (A), (B), (C), is obvious, and 
we shall now show that this analogy continues uninterrupted ; that is, 
generally, if 

du 



d n ~ l u 



then 



dx 71 ' 1 dif 



d n u _ dy 

dx* ~ dv n ~ l 



(M), 



(N). 



For considering, as before, the function Z n ~ l — - to be the differentia 

dy 

coefficient of some new function of z, viz. «„_„ so that 
we have, by differentiating (M), 



246 NOTES. 

^ . du n _, 

d n u _ dx 

dx" df 1 - 1 ' 

but the equation (A) subsisting for every function of z subsists for 
u n _ x therefore 

du n _ x _ z du n _ x ^ 
dx dy 

hence (5) 

d n u _ dy w ' dy 

dx n dif- 1 dy n ~ l 

Let now x = in the original function, and in each of the coefficients 
dn d z u d n u 

dx* dx 2 ' ' dx 2 

[«] = Y:Fy = cpy, [>] = Fy .%/[*] - fi¥y = ±y 
and 



J.37-.' -7^, then we have 



Consequently, by Maclaurin's theorem, 



#:(+»)•*» 



+ &C. 



<% 2 1-2-3 

which is the theorem of Laplace. 

The preceding investigation is taken with some slight variation 
from the large work of Lacroix, vol. i. p. 279. 

In the particular case where 

u = "¥z and z = ?/ + ;#> 
we have 

hence the development is then 



NOTES. 247 

, r dYy x . KJUJ dy a* , D 
u = T » +*"£ —I + — ^— • W + &c - 
which corresponds with the theorem of Lagrange, given at page 89. 



Note (C),page 127. 

Suppose the function f(x + /i) fails to be developable according 
to Taylor's series for the value x =■ a, and let the true development 
be represented by 

f(a + h) =fa + A/i* + (3ha+,S + C/i«-H?+y + &c (1), 

the terms being arranged according to the powers of h. It is requir- 
ed actually to find these terms. 

Let the difference f(a + h) — fa be divided by such a power of 
/i, that the quotient will become neither nor co ; when h = 0, such 

a power of h can be no other than h a , or that which ought to appear 
in the first term of this difference, for if the developed difference were 
divided by a lower power of h than this, the quotient would evidently 
be 0, when h = 0, and if it were divided by a higher power, the quo- 
tient would be infinite when h = ; hence the proper divisor h a be- 
ing found, if we put h = in the quotient, the result will be simply 
A ; having thus found the true first term of the difference, let it be 
transposed to the other side, and we shall then have the difference 

/(« + *)-/« _ A = Bt g +cft g+7 + &c . 
h a 
Now the first side of this equation being known, we have, as before, 
to find that power of /i, that it may be divided by, so that the quotient 

may be neither nor infinite, when h = this power will be lv, and 
putting h = in the quotient, the result is B, and in this manner it is 
plain that all the terms of the series are to be determined. 

Let now y = fx be the equation of a plane curve, and Y = Fx the 
equation of another having a common point with the former, at which 
x = a, then 

Fa = fa ; 



248 NOTES. 

beyond this point the ordinates of the two curves for any abscissa 
{a + h) will be f(a + h) and F (a + h), and their difference D 
will be 

D =/(<*+&) — F (a + h). 
Let us now develop the function F (a + h), as we have done the 
function/ (a + &)» and we shall have 

F (a + &) = Fa + A'A a ' + B'ft a ' +/3 ' + CT^ 4 "^'""^' + &c. (2) 
and it may be shown precisely, as at page 128, that the greater the 
number of leading terms in the two developments (1), (2), are the 
same, the nearer will the developments themselves approach to iden- 
tity, so that no curve passing through the point common to the two 
former can approach so closely to either in the vicinity of that point, 
unless in the development of the ordinate the same number of leading 
terms agree with those in (1). 

Lagrange observes (Theorie des Fonciions Analyliques, p. 184,) 
that we may call contact of the first order, contact of the second or- 
der, &c. the approximation of two curves, for which the two first 
terms, the three first terms, &c. are the same in the developments of 
the functions which represent the ordinates. All other authors who 
advert to this subject make the same remark, but it is erroneous, as a 
simple instance will show. 

Let the developed ordinates be 

A + Bh 2 + Clfi + D/i 5 + &c. 

A + B/i 2 + Ch% + Mfry + &c. 
According to Lagrange the contact here would be of the second or- 
der, but by Taylor's theorem, these developments would be 

A + O/i + B/r + 0/i 3 + 0/i 4 + &c. 

A + Oh + B/i 2 + O/i 3 + O/i 4 + &c. 
and therefore, by the general principles established in Chap. II. Sect. 
II. the contact is of the fourth order, at least. If in the true develop- 
ments the term next to O/i 4 were the same in each, and the exponent 
of h a fraction between 4 and 5, while the term following differed in 
the two series, the contact might be properly said to be of the fifth 
order ; but the sign of the difference of the two developments, when 
h is negative, will obviously depend on the fractional exponents of K 
in the terms immediately beyond those which agree in the two series, 



NOTES. 249 

Note (D),page 224. 

As the process from which the expressions for a, /3, y, given in ar- 
ticle 203, are deduced, is not immediately obvious, we shall here ex- 
hibit it at length for the first of these expressions. 

From the formulas forp" and q", at article 201, we immediately 
get for p'p" + qq" the value 

(d 2 y) (dy) (dx) — (d 2 x) (dy) 2 + (d 2 z) (dz) (dx) — {d 2 x) {dzf 



(dxf 
_ (dx) \ (d 2 y) (dy) + (dh) (dz) I - 


-(d 2 x)\(dy) 2 +(dz) 2 l 


(dxf 
_ (dx) \ (ds) (d 2 s) — (d 2 x) (dx) 2 1* 


— (d 2 x) \ (ds) 2 — (dx) 2 I 



(dx)' 
= (dx) (ds) (d 2 s) — (d 2 x) (ds) 2 '_ (ds) j (dx) (d 2 s) — (d 2 x) (ds)\ 
(dxf (dxf 

Therefore, putting, for brevity, — — - instead of the denominator, in 

( (XX j 

the expression for a in article 201, we have 

, (dsf . (d 2 x) (ds) — (d 2 s) (dx) 
a ~ * + A * (dsf 

= x + r 2 — — , article 66, 
ds 

and by a similar process the expressions for /3 and y are obtained. 

The expressions a — x, (3 — y,y — z are obviously the projec- 
tions of the radius of curvature r on the axes of a?, y, z. But, if we 
represent the inclinations of the radius to the axes by X, (x, v, the ex- 
pressions for the projections will be 

r cos. X, r cos. /x, r cos. v, 
so that (206) we have, for the angles of inclination, the values 

d?x d 2 v d?z 

cos. X — r-r—, cos. ^ — r-Y~, cos. v = v-r—. 
dsr ds 2 ds 2 

By employing these expressions M. Cauchy has arrived by rather a 
novel process at the theorem of Meusnier, given at p. 181. 

* Because from 

(&)« = {dxf + {dy)* + (dzY 

we get 

(ds) (dh) = (dx) (d*x) + (dy) (d*y) + (dz( (dh); 

32 




250 NOTES, 

Let the equation of any curve surface be 
u = F ( x, y, z) = 0, 

upon which is traced any curve MGV deter- 
mined by the equation 

y = 9^ 

Tjr~ joined to the preceding. 

If through the tangent MT to this curve, and also through the nor- 
mal MN of the surface, we draw a plane, we shall be furnished with 
a normal section MG, of which the radius of curvature r, at M, will 
be some portion of the normal MN. Also the radius of curvature r' 
of the assumed curve MG', at the same point, will be some portion 
of the line MN', perpendicular to the tangent MT. 

Now, considering s to be the independent variable, we have, for 
the inclinations of r to the axes the expressions above, viz. 
, d 2 x , dry , d 2 z 
r H^' r ~ds 2 ~' V Is*' 
and the inclinations of r or of MN to the axes are (127) 
du du du 
c?# ' dy ' dz ' 
Hence, calling the angle N'MN, between the two radii, u, we have 
{Anal. Geom.) 

, . du d 2 x , du dhi du d 2 z 

cos. w =v r (— — -f- -j -ty + -j jt)* 

v dx ds 2 dy ds 2 dz ds 2 

But the equation of the surface, considered as one of the equations 
of the curve MG', gives after two successive differentiations, still re- 
garding s as the independent variable 

du d 2 x du d?y du d?z 
dx ds 2 dy ds 2 dz ds 2 
d 2 u dx 2 d 2 u dy 2 d 2 u dz* 
S dx 2 ds 2 dy 2 ds 2 dz 2 ds 2 
) d 2 u dxdy d 2 u dxdz d 2 u dydz 

dxdy ds 2 dxdz ds? dydz ds 2 ' 

Now whatever be the curve MG', provided only its tangent MT = 
x' remains unchanged, the second member of this last equation will 

remain unchanged, because the values of -r-, ~, — , which are the 

as ds as 



NOTES. 251 

-same as those of -— , -i -7-7-, remain unchanged. Therefore 
dx dx ax 

this second member being substituted in the expression for cos. u 
leads to a result of the form 

r = K cos. w, 
K being a constant expression for all the curves on the proposed sur- 
face which touch MT at the point M. Put now in this expression 
w =0, then r' becomes r, therefore 

r = K, consequently r — r cos. w, 
which result comprehends the theorem of JWeusnier, since, if the curve 
MG' is plane, its plane will coincide with N'MT, and the angle w of 
the two radii will become the angle formed by the plane N'MT of the 
oblique section with the plane NMT of the normal section passing 
through the same tangent MT. — Leroy Analyse Jlppliquie a la Ge- 
ometric, p. 268. 

We may take this opportunity of remarking that, in our investiga- 
tion of this theorem, at p. 182, it might easily have been shown, with- 
out referring to article 86, that 

dx' 2 

—7 = 1 + tan. s 6, 

dor 

because, by the right-angled triangle, 



.;•<- 


= x 2 sec 


2 & = x 2 


(1 -4- tan 




dx' 2 




ds 2 




' ~d? ~ 


1 + tan 


2 & = — 
dx 2 



Note (E),page 106. 

The erroneous doctrine adverted to at page 106 is laid down also 
by Lacroix, in his quarto treatise on the Calculus, vol. 1, p. 340, from 
whom, indeed, Mr. Jephson seems to have adopted it. The princi- 
ple as stated by Lacroix is " que la serie de Taylor devient illusoire 
pour to.ute valeur qui rend imaginaire Pun quelconque de ces terms ; 
et que cela peut arriver sans que la fonction soit elle-meme imagi- 
naire." It is very remarkable that analysts should have hitherto held 
such imperfect notions respecting the failing cases of Taylor's theo- 
rem. 



NOTES BY THE EDITOR. 



Note (A') page 15. 

As the Algebra here referred to may not be in the hands of the 
student, we shall find the differential coefficient of a logarithmic func- 
tion, by previously obtaining that of an exponential one, which is the 
course pursued by most writers on the calculus. Let 

u = a x . . . . (1), 
in which if x be increased by /i, we shall have 
u' — a*** 1 = «*X a h . 
Now in order to develop the last factor of this product, we suppose 
a == 1 + 6, in order to subject it to the influence of the Binomial 
Theorem, we shall then have 

V + &c. 
The multiplication indicated in the second number of this being 
executed, and the result ordered according to the powers of /i, repre- 
senting by sh 2 the sum of all the terms containing powers of h above 
the first, we shall then have 

b 2 6 3 6 4 

a h = 1 + h(b — — + — — — + &c.) + s/i 2 

JL o 4 

Both members of which being multiplied by a r , designating the coeffi- 
cient of h within the parentheses by c, we shall then have 

^ x a h = u' = (1 + ch + sh 2 ) a*. 
The primitive function being taken from this, leaves 

u' — u = ca x h + cfsh 2 , 
whence 

u' — u 



h 



— ca x + « a 



NOTES. 253 



which, when h = becomes 

£ = *,.... (2), 

where 

c = ^_(a- 1 f + ia- } y_ia- } Y +kc ^ 

It is thus perceived that the differential coefficient of an exponen- 
tial function, is equal to that function multiplied by a constant num- 
ber c, which is the above function of its base. We have from equa- 
tion (2), 

du = cofdx, 

and we perceive from equation (1) that log. u = x, whence d log. 
u = dx ; eliminating dx between this and the last, we have 

du = ca x d log. tf, 
and 

71 du 1 du 

a log. u = = — . — . 

ca x c u 

The differential coefficient therefore of a logarithmic function is 
equal to the differential of the function divided by the function itself, 

multiplied by the constant — , the modulus of the system whose base 

is a. The modulus of the Naperian or Hyperbolic system of Loga- 
rithms being unity, we have 

du 
a . lu = — , 
u 



lu representing the Naperian Log. of u. 



Note (B') page 21. 

The leading part of article 15, in regard to the notation relative to 
inverse functions, though very plausible, is nevertheless calculated to 
mislead the student. For in the equation x = F -1 ?/, expressing the 
function that x is of y, the direct function being y = Fa?, the symbols 
F and F -1 should not be considered as quantities or operated upon as 
such, since they here stand in place of the words a function of, the 
forms of both functions being different. 



254 notes. 

Note (C) page 65. 

Article 49 should have commenced with the equation 
y = Far, 
and though the succeeding articles are full and ample on the subject, 
it may not be amiss to present the maxima and minima characteris- 
tics of functions in less technical language. 
Remembering the note page 63, 

let u = fx 
be the proposed function to ascertain whether it admits of maxima 
or minima values ; and if so, by what means they and the variable on 
which they depend may be discovered. 

In the proposed function if the variable x first increase and then 
decrease by any quantity h, we shall then have 

u =fx .... (1), 
and by Taylor's Theorem, 

. ,. . . , . . du h , d 2 u h 2 . d 3 u h 3 , d 4 u 

tt =/( a? + fe) = tt + _+_ 1 ^ + __^ ; +_ 

+ &c (2), 



1.2.3. 4 



du h d 2 u h 2 d?u h 3 d 4 u 

-M x — h >) - u ~di \^"d? T72~~dx T TT2Ts^~d? 

h 4 

+ &c (3). 



1.2.3.4 

Now, in order that the function u under consideration may attain a 
maximum or minimum value (2) and (3), must be both less, or both 
greater than (1), and as h may be assumed so small that the term 
containing its first power may be greater than the sum of all the suc- 
ceeding terms, (2) will be greater than u, while (3) will be less. 
Since the first differential coefficient has different signs in the two 
developments, the function therefore cannot attain maxima or mini- 
ma values, unless this coefficient becomes zero. The roots of the 

equation j- = 0, will give such values of x as may render the func- 
tion a maximum or a minimum ; such values of the variable being 



NOTES. 255 

d 2 u 
substituted in the second differential coefficient — - if these render its 

dxr 

value any thing, we are certain the function may become a maxi- 
mum if that value is negative, or a minimum if it be positive ; for in 
the first case (2) and (3) are both less than u, and in the second they 
are both greater. But if the same values of x render the second dif- 
ferential coefficient zero, as well as the first, we readily see that the 
third differential coefficient, must also become zero, in order that the 
function may admit of maxima or minima values : because this coef- 
ficient has different signs in (2) and (3), we then substitute the same 
values of x in the fourth differential coefficient, which has the same 
sign in (2) and (3), if these render it negative we shall have a maximum 
value of the function, and if positive a minimum value; but should 
this coefficient also vanish with the preceding ones, the next must be 
examined, and so on. 

In order therefore to determine the values of x, which render the 
proposed function a maximum or minimum ive must find the roots of 

the equation — = 0, and substituted then in the succeeding differen- 
tial coefficients, until we find one that does not vanish ; if this be of an 
odd order, the roots we have employed ivill not render the function a 
maximum or minimum, but if it be of an even order, then if this coeffi- 
cient be negative we have a maximum value of the function, but if posi*> 
iive a minimum value. 



THE END, 



